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# Piecewise Linear Transformation

## Key Concepts

• Translate step functions.
• Do vertical translations of the absolute value function.
• Do horizontal translations of the absolute value function.
• Understand vertical and horizontal translations.
• Understand vertical stretches and compressions.
• Understand transformations of the absolute value function.

### Piecewise Linear Transformation

#### Translate Step Functions: Piecewise Linear Transformation

Uptown Sandwich Shop is increasing the number of bonus points by 2 in the shop’s rewards program. How will the total points awarded for a \$3.80 item change?

You can represent the two versions of the reward program with step functions. Use the INT function on a graphing calculator to graph the two functions. INT is another name for floor function.

The graph is translated up by 2 units. The points for a \$3.80 item increase from 8 to 10.

### Vertical and Horizontal Translations

#### 1. Vertical Translations of the Absolute Value Function

How does adding a constant to the output affect the graph of f(x) = |x|?

Compare the graphs of h(x) = |x| – 4 and g(x) = |x| + 2 with the graph of f(x) = |x|.

Adding a constant, k, outside of the absolute value bars changes the value of f(x), or the output.
It does not change the input. The value of k, in g(x) = |x| + k, translates the graph of f(x) = |x| vertically by k units. The axis of symmetry does not change.

#### 2. Horizontal Translations of the Absolute Value Function

How does adding a constant to the input affect the graph of f(x) = |x|?

Compare the graphs of g(x) = |x – 4| with the graph of f(x) = |x|.

Adding a constant, h, inside the absolute value bars changes the value of x, the input, as well as the value of f(x), the output.

The value of h, in g(x) = |x – h| translates the graph of f(x) = |x| horizontally by h units. If h > 0, the translation is to the right. If h < 0, the translation is to the left. Because the input is changed, the translation is horizontal, and the axis of symmetry also shifts.

### Translations, Stretches and Compressions

#### 1. Understand Vertical and Horizontal Translations

What information do constants h and k provide about the graph of g(x) = |x – h| + k?

Compare the graphs of g(x) = |x – 4| – 2 and g(x) = |x + 5| + 1 with the graph of f(x) = |x|.

The value of h translates the graph horizontally and the value of k translates it vertically. The vertex of the graph

g(x) = |x – h| + k is at (h, k).

#### 2. Understand Vertical Stretches and Compressions

How does the constant a affect the graph of g(x) = a|x|?

Compare the graphs of g(x) = ½ |x| and g(x) = –4|x| with the graph of f(x) = |x|.

In g(x) = a|x|, the constant a multiplies the output of the function f(x) = |x| by a

• When 0 < |a| < 1 the graph of g(x) = a|x| is a vertical compression towards the x-axis of the graph of f(x) = |x|.
• When |a| > 1, the graph of g(x) = a|x| is a vertical stretch away from the x-axis of the graph of f(x) = |x|.
• When a < 0, the graph of g is reflected across the x-axis.

The value of a stretches or compresses the graph vertically.

### Understand Transformations of the Absolute Value Function

A. How do the constants a, h, and k affect the graph of g(x) = a|x – h| + k?

Graph g(x) = -2|x + 3| + 4.

The value of h and k determine the location of the vertex and the axis of symmetry. The value of a determines the direction of the graph and whether it is a vertical stretch or compression of the graph of f(x) = |x|.

Since |a| > 1 and a is negative the graph is a vertical stretch of the graph of f(x) = |x| that is reflected across the x-axis.

B. How can you use the constants a, h, and k to write a function whose graph is as shown below??

Step 1:

Identify the vertex of the graph.

The vertex is (4, 1), so h = 4 and k = 1.

The function has the form

f(x) = a|x – 4| + 1.

Step 2:

Find the value of a. Select another point on the graph, (x, f(x)), and solve for a.

The graph represents the function f(x) = 3|x – 4| + 1.

### Questions

Question 1

In the example about Uptown Sandwich in section 1, how will the total points be awarded for a \$1.25 juice drink change if the bonus points are decreased by 2 points?

Solution:

f(x) = Reward points and x = Dollars spent

Before: f(x) = INT(x) + 5

After: f(x) = INT(x) + 3

Before: f(1.25) = INT(1.25) + 5 = 1 + 5 = 6

After: f(1.25) = INT(1.25) + 3 = 1 + 3 = 4

The graph is translated down 2 units. The points for a \$1.25 item decrease from 6 to 4.

Question 2

For each function, identify the vertex and axis of symmetry.

1. p(x) = |x| – 2
1. p(x) = |x + 5|

Solution:

1. p(x) = |x| – 2

Here, the value of output changes and it decreases by 2 units.

The graph translates down vertically by 2 units.

Vertex: (0, -2)

Axis of symmetry remains the same, i.e., y-axis.

1. p(x) = |x + 5|

Here the value of h is -5, which is less than 0. So the graph moves horizontally to the left by
5 units.

Vertex is (-5, 0) and the axis of symmetry is x = -5.

Question 3

Find the vertex of the graph of g(x) = |x – 1| – 3

Solution:

f(x) = |x – h| + k

f(x) = |x – 1| – 3

h = 1, k = –3,

Vertex is (h, k) i.e. (1, –3)

Question 4

1. Write a function of the graph shown.
2. Write the function of the graph after a translation 1 unit right and 4 units up.

Solution:

a.

f(x) = a|x – h| + k

Vertex is (-1, 2)

So, h = -1 and k = 2

f(x) = a|x + 1| + 2

(1, 5) lies on the graph. So let’s find the value of a.

5 = aI1 + 1I + 2 …. 2a = 3… a = 1.5

Function: f(x) = 1.5|x + 1| + 2

b.

4 units up means f(x) increases by 4 and 1 unit right means h > 0 and h is 1.

f(x) = 1.5|x + 1 – 1| + 2 + 4

New Function: f(x) = 1.5|x| + 6

## Exercise

For each function, identify the vertex and the axis of symmetry.

1. f(x) = |x + 5|
2. f(x) = |x| – 2
3. f(x) = 3|x – 2| + 4
4. f(x) = 0.5|x|
5. f(x) = |x + 4|
6. f(x) = |x – 5| – 2
7. f(x) = |x + 6|
8. f(x) = 2|x|
9. f(x) = |x| + 2
10. f(x) = 2|x – 2| + 3

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