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Unit 4 hw 7 - direct variation & linear equation give 2 points

L
Lori Rapp

This document discusses direct variation, which is a linear equation that passes through the origin. It defines direct variation as y=kx, where k is the constant of variation. It provides examples of graphs that do and do not represent direct variations. It also shows step-by-step processes for finding the direct variation equation from two points, and for solving a direct variation problem when given a point and asked to find the corresponding y-value for a different x-value.

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Functions Unit 4 - Homework 7 Homework Help
What is direct variation?
What is direct variation? • A linear equation that goes through the origin.
What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b.
What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx.
What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx. • Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation.
What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx. • Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation. • General equation for direct variation is y = kx.
Which graphs are direct variations?
Which graphs are direct variations? Yes. Goes through origin.
Which graphs are direct variations? Yes. Goes through origin. No. Does NOT go through origin. Y-intercept something other than 0.
Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin. No. Does NOT go through origin. Y-intercept something other than 0.
Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin. No. Does NOT go through origin. No. Does NOT go through origin. Y-intercept something other than 0. Y-intercept something other than 0.
Steps to find Direct Variation Find the direct variation equation of the graph through the points (0, 0) and (3, -5).  Write in y=kx form.
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5).
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). • Substitute the point into y = kx.
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k.
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 − =k 3
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx.
Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx. 5 y=− x 3
You try... Find the direct variation equation of the graph through the points (0, 0) and (12, 2).  Write in y=kx form.
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form.
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx.
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx 2 = k ⋅12
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12 1 =k 6
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6
You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6 1 y= x 6
Find Direct Variation w/o point Write a direct variation equation that relates x to y.  Then solve.  Show both the equation and the solution.  If y = 15 when x = 3, find y when x = 4.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. • Substitute the “if...when” values into y = kx.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 5=k
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. • In this case we are given x=4.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. • Simplify to find y.
Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. y = 20 • Simplify to find y.
Your turn... Write a direct variation equation that relates x to y.  Then solve.  Show both the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” values into y = kx.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. 21 = k ⋅ 7
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7 3= k
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k y = 3x
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. • In this case we are given x=6.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. • Simplify to find y.
Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. y = 18 • Simplify to find y.
Write an equation in slope-intercept form from graph.
Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only!
Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points.
Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. y2 − y1 m= x2 − x1
Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1 • Substitute.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 m= = x2 − x1 0 − ( −8 ) • Substitute.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 m= = x2 − x1 0 − ( −8 ) • Substitute. • Simplify.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- 1 intercept to write y=− x−4 equation. 2
Your turn to write the equation...
Your turn to write the equation... • Identify 2 points on the graph. Use Integer coordinates only!
Your turn to write the equation... • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points.
Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. y2 − y1 m= x2 − x1
Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1 • Substitute.
Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 m= = x2 − x1 0 − ( −5 ) • Substitute.
Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 m= = x2 − x1 0 − ( −5 ) • Substitute. • Simplify.
Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify.
Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. y = mx + b • Use the slope and y- 4 intercept to write y= x+4 equation. 5
A couple comments about picking points on a Graph...
A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.)
A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.) • Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to find.
A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.) • Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to find. • Try to use the x- and y-intercepts as your points.
Write the equation given 2 points Write an equation in slope intercept form of the line that passes through (1, 2) and (4, -5).
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5).
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). y2 − y1 m= x2 − x1
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1 • Substitute and simplify.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 Will get the same slope. m= = x2 − x1 4 −1 • Substitute and simplify.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 • Solve for b.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 13 =b 3
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b 7 13 to write the equation in slope- 3 y=− x+ intercept form. 3 3
You try... Write an equation in slope intercept form of the line that passes through (-3, 7) and (2, 4).
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4).
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). y2 − y1 m= x2 − x1
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1 • Substitute and simplify.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 Will get the same slope. m= = x2 − x1 −3 − 2 • Substitute and simplify.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 • Solve for b.
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 4+ = − +b+ 5 5 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 4+ = − +b+ 5 5 5 26 =b 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 3 26 intercept form. 26 y=− x+ =b 5 5 5
What does slope mean?
What does slope mean? • It measures the steepness of a line.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means. • Reduced the slope is -1 gallon/50 miles.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement. • Reduced the slope is -1 gallon/50 miles.
What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement. • Reduced the slope is -1 gallon/50 miles. • For every 50 miles traveled one gallon of gas is used.
You try... Image from http://www.algebra-class.com/rate-of-change.html
You try... What does the slope represent in the graph to the right? Image from http://www.algebra-class.com/rate-of-change.html
You try... What does the slope represent in the graph to the right? • John’s savings account balance increase $100 each month. OR Image from http://www.algebra-class.com/rate-of-change.html
You try... What does the slope represent in the graph to the right? • John’s savings account balance increase $100 each month. OR • Every one month, John’s savings account balance increases by $100. Image from http://www.algebra-class.com/rate-of-change.html

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Unit 4 hw 7 - direct variation & linear equation give 2 points

  • 1. Functions Unit 4 - Homework 7 Homework Help
  • 2. What is direct variation?
  • 3. What is direct variation? • A linear equation that goes through the origin.
  • 4. What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b.
  • 5. What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx.
  • 6. What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx. • Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation.
  • 7. What is direct variation? • A linear equation that goes through the origin. • Remember a linear equation is in the form y = mx + b. • A direct variation would be y = mx + 0 or just y = mx. • Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation. • General equation for direct variation is y = kx.
  • 8. Which graphs are direct variations?
  • 9. Which graphs are direct variations? Yes. Goes through origin.
  • 10. Which graphs are direct variations? Yes. Goes through origin. No. Does NOT go through origin. Y-intercept something other than 0.
  • 11. Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin. No. Does NOT go through origin. Y-intercept something other than 0.
  • 12. Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin. No. Does NOT go through origin. No. Does NOT go through origin. Y-intercept something other than 0. Y-intercept something other than 0.
  • 13. Steps to find Direct Variation Find the direct variation equation of the graph through the points (0, 0) and (3, -5).  Write in y=kx form.
  • 14. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5).
  • 15. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). • Substitute the point into y = kx.
  • 16. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3
  • 17. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k.
  • 18. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3
  • 19. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 − =k 3
  • 20. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx.
  • 21. Steps to find Direct Variation Find the direct variation equation of • Need one point other the graph through the points (0, 0) and than (0, 0). Here we will (3, -5).  Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx. 5 y=− x 3
  • 22. You try... Find the direct variation equation of the graph through the points (0, 0) and (12, 2).  Write in y=kx form.
  • 23. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form.
  • 24. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx.
  • 25. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx 2 = k ⋅12
  • 26. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12
  • 27. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12
  • 28. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12 1 =k 6
  • 29. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6
  • 30. You try... Find the direct variation equation of • Use (12, 2). the graph through the points (0, 0) and (12, 2).  Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6 1 y= x 6
  • 31. Find Direct Variation w/o point Write a direct variation equation that relates x to y.  Then solve.  Show both the equation and the solution.  If y = 15 when x = 3, find y when x = 4.
  • 32. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4.
  • 33. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4.
  • 34. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. • Substitute the “if...when” values into y = kx.
  • 35. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3
  • 36. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k.
  • 37. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3
  • 38. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 5=k
  • 39. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k.
  • 40. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x
  • 41. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value.
  • 42. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. • In this case we are given x=4.
  • 43. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4.
  • 44. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. • Simplify to find y.
  • 45. Find Direct Variation w/o point Write a direct variation equation that • The “if..when” has the pieces relates x to y.  Then solve.  Show both found in an ordered pair. Use the equation and the solution.  If these values to find k. y = 15 when x = 3, find y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “find...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. y = 20 • Simplify to find y.
  • 46. Your turn... Write a direct variation equation that relates x to y.  Then solve.  Show both the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
  • 47. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
  • 48. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6.
  • 49. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” values into y = kx.
  • 50. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. 21 = k ⋅ 7
  • 51. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7
  • 52. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7
  • 53. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7 3= k
  • 54. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k
  • 55. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k y = 3x
  • 56. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value.
  • 57. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. • In this case we are given x=6.
  • 58. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6.
  • 59. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. • Simplify to find y.
  • 60. Your turn... Write a direct variation equation that • Identify “if..when” values to relates x to y.  Then solve.  Show both find k. the equation and the solution.  If y = 21 when x = 7, find y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “find...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. y = 18 • Simplify to find y.
  • 61. Write an equation in slope-intercept form from graph.
  • 62. Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only!
  • 63. Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
  • 64. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
  • 65. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points.
  • 66. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. y2 − y1 m= x2 − x1
  • 67. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
  • 68. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
  • 69. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1 • Substitute.
  • 70. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 m= = x2 − x1 0 − ( −8 ) • Substitute.
  • 71. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 m= = x2 − x1 0 − ( −8 ) • Substitute. • Simplify.
  • 72. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify.
  • 73. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
  • 74. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
  • 75. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
  • 76. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1 m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- 1 intercept to write y=− x−4 equation. 2
  • 77. Your turn to write the equation...
  • 78. Your turn to write the equation... • Identify 2 points on the graph. Use Integer coordinates only!
  • 79. Your turn to write the equation... • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
  • 80. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
  • 81. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points.
  • 82. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. y2 − y1 m= x2 − x1
  • 83. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
  • 84. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1
  • 85. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 m= x2 − x1 • Substitute.
  • 86. Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 m= = x2 − x1 0 − ( −5 ) • Substitute.
  • 87. Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 m= = x2 − x1 0 − ( −5 ) • Substitute. • Simplify.
  • 88. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify.
  • 89. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
  • 90. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4 m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. y = mx + b • Use the slope and y- 4 intercept to write y= x+4 equation. 5
  • 91. A couple comments about picking points on a Graph...
  • 92. A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.)
  • 93. A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.) • Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to find.
  • 94. A couple comments about picking points on a Graph... • Only use Integer coordinates. (No fractions or decimals.) • Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to find. • Try to use the x- and y-intercepts as your points.
  • 95. Write the equation given 2 points Write an equation in slope intercept form of the line that passes through (1, 2) and (4, -5).
  • 96. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5).
  • 97. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). y2 − y1 m= x2 − x1
  • 98. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 99. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 100. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 101. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1 • Substitute and simplify.
  • 102. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 Will get the same slope. m= = x2 − x1 4 −1 • Substitute and simplify.
  • 103. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify.
  • 104. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
  • 105. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
  • 106. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3
  • 107. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 • Solve for b.
  • 108. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3
  • 109. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 13 =b 3
  • 110. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
  • 111. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
  • 112. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
  • 113. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s. ( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to find y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b 7 13 to write the equation in slope- 3 y=− x+ intercept form. 3 3
  • 114. You try... Write an equation in slope intercept form of the line that passes through (-3, 7) and (2, 4).
  • 115. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4).
  • 116. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). y2 − y1 m= x2 − x1
  • 117. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 118. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 119. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
  • 120. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1 • Substitute and simplify.
  • 121. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 Will get the same slope. m= = x2 − x1 −3 − 2 • Substitute and simplify.
  • 122. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify.
  • 123. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
  • 124. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to find y-intercept. Choose the “easier” point to work with.
  • 125. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5
  • 126. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 • Solve for b.
  • 127. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5
  • 128. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 4+ = − +b+ 5 5 5
  • 129. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 4+ = − +b+ 5 5 5 26 =b 5
  • 130. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
  • 131. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
  • 132. You try... Write an equation in slope intercept • Find the slope. form of the line that passes through (-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope. m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 find y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept 4+ = − +b+ 5 5 5 to write the equation in slope- 3 26 intercept form. 26 y=− x+ =b 5 5 5
  • 133. What does slope mean?
  • 134. What does slope mean? • It measures the steepness of a line.
  • 135. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change.
  • 136. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run.
  • 137. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
  • 138. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
  • 139. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
  • 140. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means.
  • 141. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means. • Reduced the slope is -1 gallon/50 miles.
  • 142. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement. • Reduced the slope is -1 gallon/50 miles.
  • 143. What does slope mean? • It measures the steepness of a line. • Also referred to as rate of change. • Slope is the ratio rise/run. • To find the “meaning” of slope, identify the rise and run paying attention to the units. • Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. • The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm • So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement. • Reduced the slope is -1 gallon/50 miles. • For every 50 miles traveled one gallon of gas is used.
  • 144. You try... Image from http://www.algebra-class.com/rate-of-change.html
  • 145. You try... What does the slope represent in the graph to the right? Image from http://www.algebra-class.com/rate-of-change.html
  • 146. You try... What does the slope represent in the graph to the right? • John’s savings account balance increase $100 each month. OR Image from http://www.algebra-class.com/rate-of-change.html
  • 147. You try... What does the slope represent in the graph to the right? • John’s savings account balance increase $100 each month. OR • Every one month, John’s savings account balance increases by $100. Image from http://www.algebra-class.com/rate-of-change.html

Editor's Notes