Linear Functions

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

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

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 M=2/3 and b=-3

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

Find the slope: m 7 5

1 2

2

3Use point-slope form:

y 5 2

3x 2

Change to slope-intercept form: 3

5

3

2 xy

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 15x 5

5,0 is the

-interceptx

3x 5y 15155)0(3 y155 y3y

(0, 3) is the y-intercept

DO NOW: Find the intercepts and graph the line

1) 3x + 9y = -9

2) 4x – 2y = 16

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

DO NOW: Write an equation that matches the graph.

1) 2)

DO NOW: Write an equation that matches the graph.

1) 2)

14

1 xy 1

3

2 xy

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

2

14

2

1 xy

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

2

11

2

1 xy

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

3

8

3

4

xy