Linear Functions

Vincent J MottoUniversity of Hartford

Review of Formulas

Formula for Slope m y2 y1

x2 x1

Ax By C

y mx b

y y1 m x x1

Standard Form

Slope-intercept Form

Point-Slope Form

Find the slope of a line through points (3, 4) and (-1, 6).

m 6 4

1 3

2

4

1

2

Change into standard form.

y 3

4x 2

4 y 4 3

4x 4 2

3x 4y 8

3x 4y 8

Change into slope-intercept form and identify the slope and y-intercept.

3x 5y 15

5y 3x 15 5y

5

3x

5

15

53

35

y x

Write an equation for the line that passes through (-2, 5) and (1, 7):

Find the slope: m 7 5

1 2

2

3

Use point-slope form: y 5

2

3x 2

x-intercepts and y-interceptsThe intercept is the point(s) where the graph crosses the axis.

To find an intercept, set the other variable equal to zero. For example3x 5y 15

3x 5 0 153x 15

x 5 5,0 is the

-interceptx

Horizontal Lines Slope is zero. Equation form is y = #.

Write an equation of a line and graph it with zero slope and y-intercept of -2. y = -2Write an equation of a line and graph it that passes through (2, 4) and (-3, 4).

y = 4

Vertical Lines Slope is undefined. Equation form is x = #.

Write an equation of a line and graph it with undefined slope and passes through (1, 0). x = 1Write an equation of a line that passes through (3, 5) and (3, -2).

x = 3

Graphing Lines

*You need at least 2 points to graph a line.

Using x and y intercepts: •Find the x and y intercepts•Plot the points•Draw your line

Graph using x and y intercepts 2x – 3y = -12

x-intercept2x = -12

x = -6(-6, 0)

y-intercept-3y = -12

y = 4(0, 4)

6

4

2

-2

-10 -5

B: (0, 4)

A: (-6, 0)

B

A

Graph using x and y intercepts 6x + 9y = 18

x-intercept6x = 18

x = 3(3, 0)

y-intercept9y = 18

y = 2(0, 2)

4

2

-2

5

D: (0, 2)

C: (3, 0)

D

C

Graphing Lines

Using slope-intercept form y = mx + b:

•Change the equation to y = mx + b.•Plot the y-intercept.•Use the numerator of the slope to count the corresponding number of spaces up/down.•Use the denominator of the slope to count the corresponding number of spaces left/right.•Draw your line.

Graph using slope-intercept form y = -4x + 1:

Slopem = -4 = -4

1

y-intercept(0, 1)

4

2

-2

-4

5

F: (1, -3)

E: (0, 1)

F

E

Graph using slope-intercept form 3x - 4y = 8

Slopem = 3

4 y-intercept(0, -2)

y = 3x - 2 4

4

2

-2

-4

5

H: (0, -2)

G: (4, 1)G

H

Parallel Lines

**Parallel lines have the same slopes.

•Find the slope of the original line.•Use that slope to graph your new line and to write the equation of your new line.

Graph a line parallel to the given line and through point (0, -1):

2

-2

-4

5

3

5

Slope = 3 5

Write the equation of a line parallel to 2x – 4y = 8 and containing (-1, 4):

– 4y = - 2x + 8y = 1x - 2 2 Slope = 1

2

y - 4 = 1(x + 1) 2

y y1 m x x1

Perpendicular Lines**Perpendicular lines have the opposite reciprocal slopes.

•Find the slope of the original line.•Change the sign and invert the numerator and denominator of the slope.•Use that slope to graph your new line and to write the equation of your new line.

4

2

-2

5

-3

4

Graph a line perpendicular to the given line and through point (1, 0):

Slope =-3 4

Perpendicular

Slope= 43

Write the equation of a line perpendicular to

y = -2x + 3 and containing (3, 7):

Original Slope= -2

y - 7 = 1(x - 3) 2

y y1 m x x1 Perpendicular Slope = 1

2

Slope= 3 4

y - 4 = -4(x + 1) 3

y y1 m x x1

Perpendicular Slope = -4 3

Write the equation of a line perpendicular to

3x – 4y = 8 and containing (-1, 4):

-4y = -3x + 83

24

y x

Using Your Calculator The TI-89 Calculator is preferred. Follow the link below to acquire the skill of graphing.

Graphing with the TI-89 Graphing with the Ti-83/84

SlopeRate of Change

2 1 2 1

2 1 2 1

( ) ( )y y y f x f xm

x x x x x

Average Change

• Defined equations in terms of their changes– Linear Equations -> constant change– Exponential constant percentage change

• Will use this concept motivate derivatives

Example 4The problem: Enrique earns $6.00 per hour working

at Quikee Mart. He is saving his wages to buy a 3GB iPOD. The iPOD is on sale for $210.00. How many hours must Enrique work so he will have enough money to buy his iPOD?

• Step 1 - Make a table

• Step 2Figure how much the domain and range values are

changing. For the domain, you may use multiples of 5 to help you find the number of hours he needs to work

How do I know that this rate of change is constant?

• The ANSWER –Enrique must work 35 hours to earn his iPOD.

Make a table to help find the solution

Hours worked

Amount earned

5

10

15

20

25

30

35

+5

+5

+5

+5

+5

+30

+30

+30

+30

+30

+5 +30

$30

$60

$90

$120

$150

$180

$210

Average Change

• Rate of change– Difference between two values

• Percentage change– Difference between two values as a percentage of

the original value

• Average change– Change per unit of time

Average ChangeLet the table below describe the function r(t) = pool sales

Month Sales

1 2

2 3

3 6

4 10

5 15

6 21

7 18

8 15

9 9

10 7

11 3

12 1

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Average Change

• Rate of change

• Percent change

• Average change (slope)

)0()( rtr

)0(

)0()(

r

rtr

0

)0()(

t

rtr

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Rate of change

Rate of change from April to August

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

r(8) - r(4)

Rate of change from April to August

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

r(8) - r(4)

Average rate of change from April to August

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Average rate of change from April to August

r(8) - r(4)

8-4

Average Change

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Average rate of change from April to August

r(8) - r(4)

8-4