Use Absolute Value Functions and Transformations

Objectives:

1. To evaluate, write, and graph piecewise functions

2. To graph an absolute value function by performing SRT transformations on the parent

3. To apply SRT transformations to graphing any function

Objective 1You will be able to

evaluate, write, and graph piecewise

functions

Exercise 1

Determine whether the graph shown represents a function.

Piecewise Functions

A piecewise function is defined by more than one equation. Each equation corresponds to a different part of the domain of the function.

1 if,3

12 if,1

2 if,12

3

)(

x

xx

xx

xf

Exercise 2

Evaluate g(x) at the values below.

1. g(1)

2. g(5)

3. g(−3)

1 if,13

1 if,12)(

xx

xxxg

Exercise 3

Graph g(x).

1 if,13

1 if,12)(

xx

xxxg

Graphing Piecewise Functions

Method 1:

1. Rather than starting at the -intercept, start at the domain’s breaking point. Use the slope to graph the partial line in the correct direction.

2. Repeat for each piece of your function.

Graphing Piecewise Functions

Method 2:

1. Graph one of the equations in the piecewise function as you normally would.

2. Erase the part of the graph that you don’t need according to the domain of the piece.

3. Repeat for each piece of your function.

Exercise 4

Graph

Exercise 4

Write a piecewise function for the graph shown.

Parent Function:

The simplest member of a family of functions

Parent Functions

Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.

Family of Linear Functions Linear Parent Function

Parent Functions

Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.

Family of Quadratic Functions Quadratic Parent Function

Parent Functions

Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.

Family of Absolute Value Functions Absolute Value Parent Function

Parent Functions

Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.

Family of Functions Parent Function

A group of functions that share common characteristics

Simplest member of the family

Building the Absolute Value Function

The absolute value function is defined by .

The graph of the absolute value function is similar to the linear parent function, except it must always be positive.

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

Building the Absolute Value Function

The absolute value function is defined by .

So we just take the negative portion of the graph and reflect it across the -axis making that part positive.

Building the Absolute Value Function

The absolute value function is defined by .

This is the absolute value

parent function.

4

2

-2

-4

-5 5

f x = x

Parent Function

V-ShapeThe vertex is the minimum pointSymmetric

about the -axis

Objective 2

You will be able to graph absolute value functions using SRT transformations on the parent function.

Objective 2a

You will be able to perform vertical and horizontal shifts on the graph of a function

Translation

A translation is a transformation that

shifts a graph horizontally or vertically, but

doesn’t change the overall shape

or orientation.

Translation

The graph of

is the graph of translated horizontal units and vertical units.

The new vertex is at

Yo

u w

ill be

able to vertically

scale the

grap

h o

f

a fu

nction

Objective 2b

Stretching and Shrinking

The graph of is graph of vertically stretched or shrunk depending on the .

The value of “” gets multiplied by each -value.

For For

• The graph is stretched vertically• The graph of is narrower than

the graph of

• The graph is shrunk vertically• The graph of is wider than

the graph of

Objective 2c

You will be able to reflect the graph of a function across the x-axis

Reflection

The graph of is graph of reflected across the -axis when .

4

2

-2

-4

-5 5

f x = x

4

2

-2

-4

-5 5

f x = - x

Minuses and Pluses

You might be wondering why the ’s get a minus sign while the ’s get a plus sign. That’s just because the is on the wrong side of the equation.

𝑦=𝑎∨𝑥−h∨+𝒌 𝑦−𝒌=𝑎∨𝑥− h∨¿

𝑦− 𝑦1=𝑚 (𝑥−𝑥1 )Just think of the Point-Slope Form:

Minuses and Pluses

You might be wondering why the ’s get a minus sign while the ’s get a plus sign. That’s just because the is on the wrong side of the equation.

𝑦=3∨𝑥− 2∨+𝟓 𝑦−𝟓=3∨𝑥− 2∨¿

𝑦−7=6 (𝑥− 4 )Just think of the Point-Slope Form:

Just remember that your ’s always lie!

Objective 2

You will be able to graph absolute value functions using SRT transformations on the parent function.

Multiple Transformations

Here are two methods that you could use to graph and absolute value function. The first method will only work for absolute value functions. The second is more general and will work for any function.

In general, the graph of an absolute value function of the form can involve translations,

reflections, stretches or shrinks.

Step 4Step 3Step 2Step 1

Graphing Absolute Value Functions

Method 1: Vertex Method

Graphing is easy:

Plot vertex

Use value

as “slope” to plot

another point

Use symmetry

to find reflected

point

Connect the dots in a V-shape

Exercise 6

Without a graphing calculator, graph the following functions. How do they compare to the parent function?

1.

2.

Exercise 6

1. 2.

Objective 3 You will be able to use SRT transformations to graph any function

Multiple Transformations: SRT

Method 2: SRT Transformations

To graph , start with points on the parent function :

S R TScaling: Multiply -values

by

Reflecting: If is

negative, flip over -

axis

Translating: Move

left/right for ,

up/down for

Exercise 7

The graph of is shown. Sketch the graph of the given function.

1.

2.

3.

Exercise 7

1.

Exercise 7

2.

Exercise 7

3.

Use Absolute Value Functions and Transformations

Objectives:

1. To evaluate, write, and graph piecewise functions

2. To graph an absolute value function by performing SRT transformations on the parent

3. To apply SRT transformations to graphing any function