Geometry unit 3.7

 The ratio of its vertical rise to its horizontal
run.
 Steepness
Slope = m =
Vertical rise
Horizontal run

Find the slopes.
8 
 m
2
8
2
m 
-8
2
 4
4
m   2
2
2
4

 of a line containing two points with
coordinates (x1, y1) and (x2, y2) is
given by the formula
y 
y
2 1
x x
2 1
m



Slopes
y 
y
2 1
x x
2 1
m


5 
5
 
1 3

All horizontal
lines have a 0
slope
All vertical lines
have an
undefined
slope
0
  0
4
y 
y
2 1
x x
2 1
m


 

4 3

6 6
 7
  undefined
0

 Rise (upward) as you move left to right
Line slopes
up from left
to right
y
x

 Fall (downward) as you move left to right
Line slopes
down from
left to right
y
x

Find the slope using the
slope formula.
y 
y
2 1
x x
2 1
m


7

8
 

5 2
 
2 6
7

8


y 
y
2 1
x x
2 1
m


0  
1
 
4 0

1

4

1

4

 Describes how a quantity is changing over
time.
 The slope of a line can be used to
determine the Rate of Change
y

x


Change in quantity (y)
Change in time (x)

Recreation: For one manufacturer of camping equipment,
between 1990 and 2000 annual sales increased by $7.4 million
per year. In 2000, the total sales were $85.9 million. If the
sales increase at the same rate, what will be the total sales in
2010?
y 
y
2 1
x x
2 1
m



7.4
1
y 
85.9 2

2010 2000
85.9
7.4 2 
10
1

y
7.4(10) = y2 – 85.9
74.0 = y2 – 85.9
+85.9 +85.9
159.9 mill. = y2

 Slope-Intercept Form -
y = mx + b
slope y-intercept
 Point-Slope Form -
y – y1 = m(x – x1)
y-coordinate slope x-coordinate

1
2
y  x 
3
1
2
y  x 
3
1.) The equation is in slope-intercept form y = mx + b
2
3
The slope is
y-intercept (0, 1)
2.) Plot the point (0, 1)
2
3.) Use the slope , from
3
the point (0, 1) go up 2,
right 3

y  3x 1
1.) The equation is in slope-intercept form y = mx + b
1
2
y  x 
3
The slope is 3
y-intercept (0, 1)
3.) Use the slope 3, from
the point (0, 1) go up 3,
right 1

1.) The equation is in point-slope form y – y1 = m(x – x1)
The slope is -2
Point on line (-3, 3)
2.) Plot the point (-3, 3)
3.) Use the slope -2, from
the point (-3, 3) go
down 2, right 1

1
y  2   x 
( 4)
3
1.) The equation is in point-slope form y – y1 = m(x – x1)
1
 The slope is
3
Point on line (4, 2)
1

3.) Use the slope , from
3
the point (4, 2) go
down 1, right 3

 If we know the slope and at least one point
 If we have the slope and y-intercept, use the
slope-intercept form; y = mx + b
 If we have the slope and a point, use the
point-slope form; y – y1 = m(x – x1)

What is an equation of the line with slope 3
and y-intercept -5?
 Start with the slope-intercept form of
the equation
y = mx + b
y = 3x + (-5) Substitute 3 for m, and -5
for b
y = 3x - 5 Simplify

What is an equation of the line through point
(-1, 5) with slope 2?
 Start with the point-slope form of the
equation
y – y1 = m(x – x1)
y – 5 = 2(x - (-1)) Substitute 2 for m, and -1
in for x1 and 5 in for y1
y – 5 = 2(x + 1) Simplify

What is an equation of the line with
1

slope and y-intercept 2?
2
 Start with the slope-intercept form of
the equation
y = mx + b
1
1

y = x + 2 
Substitute for m, and
2 for b 2
2

What is an equation of the line through point
(-1, 4) with slope -3?
 Start with the point-slope form of the
equation
y – y1 = m(x – x1)
y – 4 = -3(x - (-1)) Substitute -3 for m, and -1
in for x1 and 4 in for y1
y – 4 = -3(x + 1) Simplify

 If we know two points on the line
 Find the slope using the formula
 Using the point-slope formula
 Plug in one of the two points
 Plug in the slope for m

What is an equation of the line through point (-2, -1)
and point (3, 5)?
y 
y
2 1
x x
2 1
m

 Find the slope 
6
y + 1 = (x + 2) or
5
6
 

5 1
 
6

6

y - 3 = (x - 5)
5
3 2
5


5
 Start with the point-slope form of the equation
y – y1 = m(x – x1)
 Plug in the slope and one of the two points

 We don’t need a slope
 All points on a horizontal line have the
same y-coordinate; so the equation is y = y1.
 All points on a vertical line have the same
x-coordinate; so the equation is x = x1.
 Where (x1, y1)

What are the equations for the horizontal and
vertical lines through (2, 4)?
 The horizontal is y = y1
y = 4 Substitute 4 for y1
 The vertical is x = x1
x = 2 Substitute 2 for x1

What are the equations for the horizontal and
vertical lines through (4, -3)?
 The horizontal is y = y1
y = -3 Substitute -3 for y1
 The vertical is x = x1
x = 4 Substitute 4 for x1

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Geometry unit 3.7

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Geometry unit 3.7