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13.4 – Slope and Rate of Change
X
Yrise
sloperun
change in yslope
change in x
Slope is a rate of change.
1 2
1 2
y yslope
x x
2 2,x y
1 1,x y
X
Ychange in yslope
change in x
1 2
1 2
y yslope
x x
1,0
4,5
slope
5
3
5 04 1
slope
13.4 – Slope and Rate of Change
X
Ychange in yslope
change in x
1 2
1 2
y yslope
x x
5,1
4,5
slope
slope4
9
5 14 5 4
9
13.4 – Slope and Rate of Change
X
Y
1 2
1 2
y yslope
x x
2, 4
2,3
slope
slope undefined
Slope of any Vertical Line
2x
3 4 2 2
7
0
13.4 – Slope and Rate of Change
X
Y
1 2
1 2
y yslope
x x
3, 3
4, 3
slope
slope 0
Slope of any Horizontal Line
3y
3 3 4 3
0
7
13.4 – Slope and Rate of Change
5 4 10x y
slope
slope
0y
5 6 4 10y
5y
5 2 4 10y
6, 5 2,0
Find the slope of the line defined by:
6x
30 4 10y 4 20y
2x
10 4 10y
4 0y
5 0 6 2
5
4
13.4 – Slope and Rate of Change
If a linear equation is solved for y, the coefficient of the x represents the slope of the line.
5 4 10x y
4 5 10y x
5 10
4 4y x
5
4
5
2y x
Alternative Method to find the slope of a line
5 4 10x y
slope5
4
13.4 – Slope and Rate of Change
2 7x y
2slope
If a linear equation is solved for y, the coefficient of the x represents the slope of the line.
5 7 2x y
5
7slope
2 7y x
2 7y x
7 5 2y x 2
7
5
7y x
13.4 – Slope and Rate of Change
5x y
1slope
Parallel Lines are two or more lines with the same slope.
2 2 3x y
1slope
3
2y x
5y x 2 2 3y x
These two lines are parallel.
13.4 – Slope and Rate of Change
5x y
2
5slope
Perpendicular Lines exist if the product of their slopes is –1.
5 2 1x y
5
2slope
5 1
2 2y x
5 2 3y x
These two lines are perpendicular.
2 3
5 5y x
2 5 1y x
2 51
5 2
13.4 – Slope and Rate of Change
3 9 5 0x y
slope
Are the following lines parallel, perpendicular or neither?
3 2x y
slope
11
3y x
9 3 5y x
NEITHER
3 5
9 9y x
3 2y x
1 5
3 9y x
1
3
1
3
13.4 – Slope and Rate of Change
6 12 4x y
slope
2 3x y
slope
12 6 4y x
These two lines are perpendicular.
6 4
12 12y x
2 3y x
1
2
Are the following lines parallel, perpendicular or neither?
1 1
2 3y x
1
2
2
2 1
13.4 – Slope and Rate of Change
For every twenty horizontal feet a road rises 3 feet. What is the grade of the road?
riseslope
run
3
20
feetslope
feet
% 100%grade slope
3% 100%
20grade
% 15%grade
13.4 – Slope and Rate of Change
The pitch of a roof is a slope. It is calculated by using the vertical rise and the horizontal run. If a run rises 7 feet for every 10 feet of horizontal distance, what is the pitch of the roof?
risepitch slope
run
7
10
feetpitch
feet
7
10pitch
13.4 – Slope and Rate of Change
13.5 – Equations of Lines
Slope-Intercept Form– requires the y-intercept and the slope of the line.
y mx b
m = slope of line b = y-intercept
24
3y x
Slope-Intercept Form:
y mx b
m = slope of line b = y-intercept
62
5y x
13.5 – Equations of Lines
Slope-Intercept Form:
y mx b
m = slope of line b = y-intercept
33
2y x
13.5 – Equations of Lines
Slope-Intercept Form:
y mx b
m = slope of line b = y-
intercept
34
4y x
13.5 – Equations of Lines
Write an equation of a line given the slope and the y-intercept.
9intercept = 2
11m y y
7intercept 5
3m y y
21 30,
13 4m
y
9
11x 2
7
3 x 5
21
13x
3
4
y mx b
13.5 – Equations of Lines
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.
1 1y y m x x
11 2
3y x
2,11
3m
13.5 – Equations of Lines
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.
1 1y y m x x
32 3
4y x
3,2 3
4m
13.5 – Equations of Lines
Point-Slope Form – requires the coordinates of a point on the line and the slope of the line.
1 1y y m x x
4
4 25
y x
2, 4 4
5m
13.5 – Equations of Lines
Writing an Equation Given Two Points
Ax By C
1. Calculate the slope of the line.
2. Select the form of the equation.
a. Standard form
b. Slope-intercept form
c. Point-slope form
y mx b
1 1y y m x x
3. Substitute and/or solve for the selected form.
13.5 – Equations of Lines
Writing an Equation Given Two Points
1,3 5, 2
3 2
1 5m
5
4
5
4
2 3
5 1m
5
4
5
4
or
Given the two ordered pairs, write the equation of the line using all three forms.
Calculate the slope.
13.5 – Equations of Lines
Writing an Equation Given Two Points
1,3 5, 25
4m
1 1y y m x x
53 1
4y x 5
2 54
y x
Point-slope form
13.5 – Equations of Lines
Writing an Equation Given Two Points
1,3 5, 2 5
4m
53 1
4y x 5
2 54
y x
5 53
4 4y x
5 53 3 3
4 4y x
5 17
4 4y x
5 252
4 4y x
5 252 2 2
4 4y x
5 17
4 4y x
Slope-intercept form
13.5 – Equations of Lines
Writing an Equation Given Two Points
1,3 5, 2 5
4m
5 17
4 4y x
Standard form
LCD: 4
5 17
4 44 4 4y x
4 5 17y x
5 4 17x y
13.5 – Equations of Lines
Solving Problems
The pool Entertainment company learned that by pricing a pool toy at $10, local sales will reach 200 a week. Lowering the price to $9 will cause sales to rise to 250 a week.
a. Assume that the relationship between sales price and number of toys sold is linear. Write an equation that describes the relationship in slope-intercept form. Use ordered pairs of the form (sales price, number sold).
b. Predict the weekly sales of the toy if the price is $7.50.
13.5 – Equations of Lines
Solving Problems
,sales price number sold
10,200 9,250
250 200
9 10m
50
1m
50m
200 50 10y x
200 50 500y x
50 700y x
250 50 9y x
250 50 450y x
50 700y x
13.5 – Equations of Lines
Solving Problems
,sales price number sold
50 700y x 7.50x
Predict the weekly sales of the toy if the price is $7.50.
50 7.50 700y
375 700y
325y items sold
13.5 – Equations of Lines