Math 2: Unit 5BDay 2

Graphing absolute value functions and

transformations

Absolute Value Parent Function

( )f x x

x

y

: ( , )Vertex h k

: AOS x h

Transformations and translations of the parent function

The graph has vertex (h,k) and is symmetric in the line x = h

h shifts the parent function horizontally (remember to change the sign)

k shifts the parent function vertically The graph is V-shaped. It opens up if a > 0 and down if a < 0. The graph is wider than the graph of if

< 1 and narrower if >1.

( )f x a x h k

y x

a a

Transformations and translations of the parent function

( ) 2 1 3f x x

: (1,3)Vertex

: 1AOS x

x

y

Identify the vertex, tell whether the graph opens up or down, then tell

whether the graph is wider or narrow or normal compared to y x

1. 2 3 5y x 22. 4

3y x

Vertex (-3,-5), opens up, Narrower.

Vertex (0,-5), opens up, Wider.

Identify the vertex, tell whether the graph opens up or down, then tell

whether the graph is wider or narrow or normal compared to y x

3. 3 5 2y x 14. 4

2y x 5. 5y x

Vertex (5,-2), opens down, Narrower.

Vertex (½,4), opens up, Normal or the same.

Vertex (5,0), opens down, Normal or the same.

To find the x-intercept(s): substitute in 0 for y To find the y-intercept: substitute in 0 for x From a graph, the intercepts are where the

graph crosses each axis.1

( ) 2 12

f x x

Intercepts

No x-intercepts!y-intercept (0,-2)

Domain (x-values): read from left to right Range (y-values): read from bottom to top

Domain and Range

Find the domain and rangeof this function.

x

y

Domain: all real numbers Range: y ≥ 2.

Domain (x-values): read from left to right Range (y-values): read from bottom to top

Domain and Range

Find the domain and rangeof this function.

x

y

Domain: all real numbers Range: y ≥ - 4 .

Domain (x-values): read from left to right Range (y-values): read from bottom to top

Domain and Range

Find the domain and rangeof this function.

Domain: all real numbers Range: y ≤ - 1 .

x

y

The values of x when f(x) = 0 To find the zeros of any function: plug in y =

0 and solve for x. Find the zero(s) of

( ) 2 1 3f x x

Zeros

x = -1 and x = 3 .

x

y

Decreases: Increases:

Find the intervals where this function is increasingand decreasing.

Intervals of increasing and decreasing

x

y

Decreases: (-∞, -3)

Increases: (-3,∞)

Decreases: Increases:

Find the intervals where this function is increasingand decreasing.

Intervals of increasing and decreasing

x

y

Decreases: (-∞, 2)

Increases: (2,∞)

Decreases: Increases:

Find the intervals where this function is increasingand decreasing.

Intervals of increasing and decreasing

Increases: (-∞, -2)

Decreases: (-2,∞)

x

y

Graphing

To graph an absolute value function– Find the vertex and sketch the AOS– Use a table of values choosing 2 x-values on

each side of the vertex– plot the points and draw the graph

Vertex: Max/min? Axis of symmetry: Domain: Range: Y-intercept: Zeros: Intervals of increase and decrease

( ) 2 1f x x

x

y

Graphing

( - 2, - 1)

minimum

x = - 2

all real numbers

y ≥ - 1 .

( 0, 1)

( -3, 0) and ( -1, 0)

Decreases: (-∞, -2)

Increases: (-2,∞)

Graph the following: ( ) 3 1f x x

x y Vertex: Max/min? AOS: Domain: Range: Intercepts: Zeros: Intervals of increase and decrease

Graph the following:

Vertex: AOS: Domain: Range: Intercepts: Zeros: Intervals of increase and decrease

( ) 2f x x

x y

Graph the following:1

( ) 1 12

f x x

x y Vertex: Max/min? AOS: Domain: Range: Intercepts: Zeros: Intervals of increase and decrease

20

Calculating average rate of change

• Do you remember how to find slope when given 2 points?

•If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing.

•Slope is the average rate of change!

On the following graph, where is the rate of change positive? Where is it negative?

x

y

Tell about the interval:- £ £ -3 1x

On the following graph, where is the rate of change positive? Where is it negative?

x

y

Tell about the interval:- £ £ -2 3x

Assignment:

Handout