Linear Functions

Compare and Contrast

Yards to Feet

Num

ber o

f Fee

t

Number of Yards

Yards to Square Yards

Length of a Side of a Square Yard

Area

of S

quar

e (y

d2 )

Tables: Linear or Nonlinear

• linear • nonlinear

x y

2 50

4 35

6 20

8 5

x y

1 1

4 16

7 49

10 100

Is the rate of change constant (the same)?

+2

+2

+2

-15

-15

-15

+3

+3

+3

+15

+33

+51

Identify: Linear or Nonlinear Table?

• linear

x y

0 20

5 16

10 12

15 8

x 0 2 4 6

y 0 2 8 18

nonlinear

Graphs: Linear or Nonlinear

Is the graph a straight line?

linear linear nonlinear

Identify: Linear or Nonlinear Graph?

linear linear nonlinear

Equations: Linear or Nonlinear

REMEMBER: x1 = x and x0 = 1

In “y = ” form, is x raised to a power of 1 or 0?

Does x appear in the numerator?

y = x + 4y = 6/x

y = 4y = x3 + 1

y = ½ xlinear

nonlinear

linear

nonlinear

linear

Identify:Linear or Nonlinear Equation?

y = 2/x + 5y = x2 + 8y = .6x1 y = + 13x2

linear linearnonlinear nonlinear

Pointers to Keep in Mind

• A graph is linear if it is a straight line.

A table is linear if the rate of change is constant. There is a common difference.

An equation is linear if the power of x is either 1 or 0 and it appears in the numerator.

Exit Slip

Identify if linear or nonlinear.1. Table A

2. Graph

3. Equation x 3 6 9 12

y 12 10 8 6

a b c

y = + 58x

Review of Formulas

Formula for Slope m y2 y1x2 x1

Ax By C

ymx b

y y1 m x x1

Standard Form

Slope-intercept Form

Point-Slope Form

*where A>0 and A, B, C are integers

Change into standard form.

y3

4x 2

4 y4 3

4x 4 2

3x 4y 8

3x 4y8

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.3x 5y 15

3x 5 0 153x 15

x 5 5,0 is the

-interceptx

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

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

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

Review of Formulas

Formula for Slope m y2 y1x2 x1

Ax By C

ymx b

y y1 m x x1

Standard Form

Slope-intercept Form

Point-Slope Form

*where A>0 and A, B, C are integers

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 35

5m and b

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

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