3.2 graphing functions and relations. another way that we can determine if a relation is a function...

3.2 Graphing Functions and Relations

• Another way that we can determine if a relation is a function is by looking at a graph of the relation.

• A relation is a function if and only if no

line intersects the graph more than once.

vertical

Graph the relation and then state whether or not the relation is a function.

1. 0,4 , 3,0 , 1, 2 , 3,1 , 4,0

Graph the relation and then state whether or not the relation is a function.

2. 1, 1 , 1,2 , 4,2 , 5, 3 , 2, 3

Graph the relation defined by each of the following rules over the domain

3.

2, 1,0,1,2, 3 y 5 x

Graph the relation defined by each of the following rules over the domain

4.

2, 1,0,1,2, 3 x y 3

3.3 Graphing Linear Equations

Linear Equations• Linear equations contain both an and a

• The graph of a linear equation is a .

• There are two forms of linear equations that we will look at:1. , which is in the form

2. , which is in the form

x y

line

Slope-intercept

Standard

y mx b

Ax By C

Slope-Intercept Form

• The m represents the of the line. The slope is the of the line. – When the slope is positive, the graph goes from left

to right.– When the slope is negative, the graph goes from

left to right.

– When graphing, the slope can be thought of as over , where the numerator determines the

displacement and the denominator determines the displacement between points on the line.

y mx bslope

steepnessup

down

riserun

verticalhorizontal

Slope-Intercept Form

• The b represents the of the line. In other words, it’s where the line crosses the

.

y mx b

y-intercept

y-axis

Determine the slope and y-intercept of the lines below.

1.

2.

y 1

2x 4

3y x 12

Graphing in Slope-Intercept Form

• First, make sure that the equation is in the form .

• Plot the first. Use this as your starting point.

• Use the slope to plot another point, using the idea of rise over run.

• Draw a line through your points with arrows on each end.

y=mx+b

y-intercept

Graph the equation.

3. y 2

3x 4

Graph the equation.

4. y 4x 7

Graphing in Standard Form

• One way to graph from standard form is to transform the equation into slope-intercept form.

• A second way to graph in standard form is to find the x-intercept and y-intercept. Plot the two points and draw a line through them.

x-intercept

• The x-intercept is in the form . So to find the x-intercept, you will need to substitute into the equation for and solve for .

, x 0

0 yx

y-intercept

• The y-intercept is in the form . So to find the x-intercept, you will need to substitute into the equation for and solve for .

• You can use this method to find the x and y-intercepts in slope-intercept form as well!

, 0 y

0 xy

Find the x-intercept and y-intercept for the following equations.

5.

6.

3x 2y 12

x

5y

61

Graph using intercepts.

7. 5x 2y 10

Graph using intercepts.

8. 6x 3y 12

9.Determine the value of k so that the given point will be on the graph of the given equation.

4x 6k 3y; 3, 10

Vertical and Horizontal Lines

• The equation for a vertical line is in the form: . The slope of a vertical line is

.

• The equation for a horizontal line is in the form: . The slope of a horizontal line is

.

x = #undefined

y = #

zero

Graph the Equation

10. y 5

Graph the Equation

11. x 3

Absolute Value Graphs

• Absolute value graphs are in the shape of a .V

Use a table of values to graph the function.

12. y x

• On your graphing calculator graph the following functions:

• Y1:

• Y2:

• Y3:

• What happened to the graph?

y xy x 3

y x 4

• On your graphing calculator graph the following functions:

• Y1:

• Y2:

• Y3:

• What happened to the graph?

y x2

y x 3 2

y x 2 2

• On your graphing calculator graph the following functions:

• Y1:

• Y2:

• Y3:

• What happened to the graph?

y x

y x 5 3y x 1 4

13. Without your calculator, graph the following functions.

a. y x 1 2

13. Without your calculator, graph the following functions.

b. y x 2 3