Linear function and slopes of a line
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This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
Linear function and slopes of a line
- 1. Linear Functions linear equations, intercepts and slopes
- 2. A linear equation is the equation of a line. The standard form of a linear equation is Ax + By = C * A has to be positive and cannot be a fraction.
- 3. Examples of linear equations 2x + 4y =8 The equation is in the standard form 6y = 3 – x x + 6y = 3 4x − y = −7 3 4x - y = 21
- 4. Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: 4x2 + y = 5 x=4 xy + x = 5 s/r + r = 3 The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided
- 5. Determine whether the equation is a linear equation, if so write it in standard form. y = 5 − 2x 2x + y = 5 y = x +3 This is not a linear equation since its in the second degree 2 2 xy = −5 y + 6 1 x + 5y = 3 4 This is not a linear equation since variables are multiplied x + 20y = 12
- 6. DEFINITION OF A LINEAR FUNCTION A linear function is a function of the form f(x) = mx + b where m and b are real numbers and m = 0
- 7. Transform the following into the form y = mx + b x +y=2 2x 8x y = -x + 2 –y=5 y = 2x - 5 – 2y = 12 y = 4x - 6 -3x + 2y = 6 y = 3x + 3 2
- 8. SLOPE OF A LINE Slope refers to the steepness of a line.
- 9. Slopes: trends An increasing line defines a positive slope A decreasing line defines a negative slope A horizontal line defines a zero slope A vertical line defines an undefined slope
- 10. Finding the slope of Linear functions What is the slope of a line passes through points (4,6) and (3,4)? m=2
- 11. Determine the Slope of the following linear functions that passes through the given pair of points 1. (3, 2), (6, 6) 2. (-9, 6), (-10, 3) 3. (-4, 2), (-5, 4)
- 12. x and y intercepts The x coordinate of the point at which the graph of an equation crosses the x –axis is the x- intercept. The y coordinate of the point at which the graph of an equation crosses the y-axis is called the y- intercept. y- intercept (0, y) X- intercept (-x,0)
- 13. Graph the linear equation using the x- intercept and the y intercept 3x + 2y = 9 To find the x- intercept, let y = 0 3x + 2y = 9 3x + 2(0) = 9 3x = 9 x=3 Replace y with 0 Divide each side by 3 To find the y- intercept, let x = 0 3x + 2y = 9 3(0) + 2y = 9 2y = 9 y = 9/2 Replace x with 0 Divide each side by 2 Plot the two points and connect them to draw the line.
- 14. 2x + y = 4 To find the x- intercept, let y = 0 2x + y = 4 Original Equation 2x + (0) = 4 2x =4 x=2 Replace y with 0 Divide each side by 3 To find the y- intercept, let x = 0 2x + y = 4 Original Equation 2(0) + y = 4 y=4 Replace x with 0 Simplify Plot the two points and connect them to draw the line.
- 15. Find the x and y- intercepts of x = 4y – 5 ● ● ● x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x=0-5 x = -5 (-5, 0) is the x-intercept ● ● y-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y 5 =y 4 5 ● (0, ) 4 is the y-intercept
- 16. Find the x and y-intercepts of g(x) = -3x – 1* ● ● x-intercept 1 ( − , 0) is the 3 x-intercept *g(x) is the same as y ● ● y-intercept (0, -1) is the y-intercept
- 17. Find the x and y-intercepts of x = 3 ● x-intercept ●There is no y. x = 3 is a vertical line so x always equals 3. ● ● ● y-intercept A vertical line never crosses the y-axis. ● ● There is no y-intercept. (3, 0) is the x-intercept. x y
- 18. Find the x and y-intercepts of y = -2 ● x-intercept Plug in y = 0. y cannot = 0 because y = -2. ● y = -2 is a horizontal line so it never crosses the x-axis. ● ●There ● y-intercept ● y = -2 is a horizontal line so y always equals -2. ● (0,-2) is the y-intercept. x is no x-intercept. y
- 19. EQUATION OF A LINEAR FUNCTION Slope- Intercept form y = mx + b
- 20. y = mx + b Give the equation of the linear function y in slope intercept form given its slope and y-intercept 1. m = -3, b = 2 2. m= 2, b = - 4 3. M = 1/3, b = 3
- 21. EQUATION OF A LINEAR FUNCTION Point-Slope form y –y1= m(x – x1)
- 22. y –y1= m(x – x1) Give the equation of the linear function y with the given slope and passing through given points. 1. m = 2, through (1, 2) 2. m= -3, through (5, 0) 3. m = -1/3, through (-1, 3)
- 23. EQUATION OF A LINEAR FUNCTION
- 24. Give the equation of the linear function y with the given slope and passing through given points. 1. through (1, 2) and (3, -2) 2. through (5, 0) and (-1, 3)
- 25. EQUATION OF A LINEAR FUNCTION Intercept Form _x_ + _y_ a b = 1
- 26. Questions??