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