Algebra 2 by Richard Wright

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0-06 Graph Absolute Value Functions and Transformations

Imagine you are creating a digital piece of art. For part of it you want to copy a shape and move it over and up, then make it larger. The moving and enlarging are transformations.

Transformations

Transformations are operations to functions that change the graph’s size, shape, position, or orientation. The basic format of transformations is h(x) = a · f(x − h) + k where h(x) is the transformed function and f(x) is the original function. There are four transformations covered in this lesson.

a determines the vertical stretch or shrink. The graph of the function is stretched or shrunk vertically by the factor of a. The y-coordinates are multiplied by a. For example, if a is 2, then the graph is twice as tall.

Reflections flip the graph over a line. If a is negative, the graph will be flipped over the x-axis.

Translations move the graph. h, which is with the x, is how far graph moves to right (x direction). k, which is with the whole function, is how far graph moves up (y direction).

Apply stretch/shrinks and reflections before translations because order of operations says to multiply before adding.

Example 1

The graph of f(x) is given. Sketch the following functions.

1. y = −2f(x)
2. \(y=\frac{1}{2}f\left(x-1\right)-3\)

Solution

1. a = −2 so the graph is reflected over the x-axis and twice as tall. Multiply the y-coordinates by −2.

   For example, the left most point was (−2, −2). Multiply its y-coordinate by −2 to get (−2, 4). Do the same thing to the other points.

   The transformations are reflected over the x-axis and stretched by a factor of 2.

   

2. \(a=\frac{1}{2}\) so the graph is half as tall. Multiply the y-coordinates by \(\frac{1}{2}\). h = 1 and k = −3 so shift the new points 1 right and 3 down.

   For example, the left point was (−2, −2). Multiply the y-coordinate by \(\frac{1}{2}\) giving (−2, −1). The move it 1 to the right and 3 down giving (−1, −4). Repeat for the other points.

   The transformations are shrunk by factor of \(\frac{1}{2}\) and shifted 1 right and 3 down.

   

Absolute Value Graph

The graph of an absolute value function looks like a V because the anything in the absolute value becomes positive. The point of the graph is called the vertex. The slope of the right side is 1 and the slope of the left side is −1.

If transformations are applied to the absolute value graph, the result is

y = a|x − h| + k

First stretch the graph vertically by a. The slope of the right side was \(1=\frac{1}{1}=\frac{y}{x}\). If the y is stretched, then the new slope will \(a\cdot\frac{1}{1}=a\).

Then shift the graph using h and k. Since the vertex was (0, 0), then the new vertex will be (0 + h, 0 + k) = (h, k).

Example 2

Describe the transformation, then graph the function.

f(x) = |x + 2| − 1

Solution

Compare the given function with y = a|x − h| + k.

a = 1 so there is no stretch. h = −2 and k = −1 so the graph is translated 2 left and 1 down.

The vertex is (h, k) = (−2, −1). The slope of the right side is a = 1.

The transformations are translated 2 left and 1 down.

Example 3

Describe the transformation, then graph the function.

f(x) = 2|x|

Solution

Compare the given function with y = a|x − h| + k.

a = 2 so there the graph is twice as tall. h = 0 and k = 0 so the graph has no translation.

The vertex is (h, k) = (0, 0) and the slope of the right side is a = 2.

The transformation is stretched by a factor of 2.

Example 4

Describe the transformation, then graph the function.

$$ f\left(x\right)=\frac{1}{2}\left|x-1\right|+2 $$

Solution

Compare the given function with y = a|x − h| + k.

\(a=\frac{1}{2}\) so there the graph is half as tall. h = 1 and k = 2 so the graph is translated 1 right and 2 up.

The vertex is (h, k) = (1, 2) and the slope of the right side is \(a = \frac{1}{2}\).

The transformations are shrunk by factor of \(\frac{1}{2}\) and translated 1 right and 2 up.

Example 5

Write an absolute value function to model the graph.

Solution

The vertex is (0, 3). Since the vertex is (h, k), h = 0 and k = 3. The slope of the right side is −3 so a = −3. Fill those values into y = a|x − h| + k.

f(x) = −3|x| + 3

Practice Exercises

The graph of f(x) is given. a) Describe the transformations. b) Sketch the graph.



1. 3f(x)
2. −f(x)
3. f(x – 1) – 3
4. f(x + 2) + 3
5. 2f(x) – 1

a) Describe the transformation, then b) graph the absolute value function.

6. \(f\left(x\right)=\frac{3}{2}\left|x\right|\)
7. g(x) = –|x|
8. h(x) = |x – 4| + 1
9. y = |x + 2| – 1
10. \(y=\frac{1}{2}\left|x+1\right|\)
11. y = –3|x – 1| – 2

Write an absolute value function to model the graph.

12. 
13. 
14. 

Mixed Review

15. (0-05) Graph y = 2x – 1.
16. (0-05) Graph 3x – y = 9.
17. (0-04) Find the equation of the line passing through (2, 4) and (4, –1).
18. (0-03) Solve \(\frac{1}{2}\left|3-x\right|=5\).
19. (0-02) The thermometer in the classroom reads 74 °F, but your friend from another country only understands temperature in °C. What is the temperature of your classroom in °C?
20. (0-01) Solve 3 < 2x + 1 ≤ 11

Answers

1. ; Vertical stretch by factor of 3.
2. ; Reflected over the x-axis.
3. ; Shift right 1 and down 3.
4. ; Shift left 2 and up 3.
5. ; Stretch by a factor of 2 and shift down 1.
6. ; Stretch by a factor of \(\frac{3}{2}\).
7. ; Reflect over the x-axis.
8. ; Shift right 4 and up 1.
9. ; Shift left 2 and down 1.
10. ; Stretch by a factor of \(\frac{1}{2}\) and shift left 1.
11. ; Reflect over x-axis, stretch by a factor of 3, and shift right 1 and down 2.
12. f(x) = |x| – 1
13. f(x) = |x + 2| – 2
14. f(x) = 2|x + 1|
15. 
16. 
17. \(y=-\frac{5}{2}x+9\)
18. −7, 13
19. 23.3 °C
20. 1 < x ≤ 5