Transformations of Absolute Value Functions

Oct 13, 2025 by windailys.com 97 views

Absolute value functions are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to model real-world problems, such as distance, speed, and temperature. In this article, we will explore the properties of absolute value functions and determine which function, when graphed, will be narrower than the graph of the parent function, $f(x) = |x|$ .

The Parent Function

The parent function of absolute value functions is $f(x) = |x|$ . This function is a V-shaped graph with its vertex at the origin (0, 0). The graph is symmetric about the y-axis, and its equation can be written as:

f(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Transformations of Absolute Value Functions

Absolute value functions can be transformed in various ways, including horizontal shifts, vertical shifts, and vertical stretches or compressions. These transformations can be achieved by adding or subtracting constants from the function, or by multiplying or dividing the function by a constant.

Horizontal Shifts

A horizontal shift of an absolute value function can be achieved by adding or subtracting a constant from the function. For example, the function $f(x) = |x + 2|$ is a horizontal shift of the parent function $f(x) = |x|$ by 2 units to the left.

Vertical Shifts

A vertical shift of an absolute value function can be achieved by adding or subtracting a constant from the function. For example, the function $f(x) = |x| - 3$ is a vertical shift of the parent function $f(x) = |x|$ by 3 units down.

Vertical Stretches and Compressions

A vertical stretch or compression of an absolute value function can be achieved by multiplying or dividing the function by a constant. For example, the function $f(x) = 0.5|x|$ is a vertical compression of the parent function $f(x) = |x|$ by a factor of 0.5.

Comparing the Functions

Now that we have discussed the transformations of absolute value functions, let's compare the given functions to determine which one will be narrower than the graph of the parent function, $f(x) = |x|$ .

A. $f(x) = |x| - 3$

This function is a vertical shift of the parent function $f(x) = |x|$ by 3 units down. Since the graph is shifted down, it will be narrower than the graph of the parent function.

B. $f(x) = |x + 2|$

This function is a horizontal shift of the parent function $f(x) = |x|$ by 2 units to the left. Since the graph is shifted to the left, it will be wider than the graph of the parent function.

C. $f(x) = 0.5|x|$

This function is a vertical compression of the parent function $f(x) = |x|$ by a factor of 0.5. Since the graph is compressed vertically, it will be narrower than the graph of the parent function.

D. $f(x) = 4|x|$

This function is a vertical stretch of the parent function $f(x) = |x|$ by a factor of 4. Since the graph is stretched vertically, it will be wider than the graph of the parent function.

Conclusion

Based on the transformations of absolute value functions, we can conclude that the function $f(x) = 0.5|x|$ will be narrower than the graph of the parent function, $f(x) = |x|$ . This is because the function is a vertical compression of the parent function by a factor of 0.5, resulting in a narrower graph.

Key Takeaways

Absolute value functions can be transformed in various ways, including horizontal shifts, vertical shifts, and vertical stretches or compressions.
Horizontal shifts can be achieved by adding or subtracting constants from the function.
Vertical shifts can be achieved by adding or subtracting constants from the function.
Vertical stretches and compressions can be achieved by multiplying or dividing the function by a constant.
The function $f(x) = 0.5|x|$ will be narrower than the graph of the parent function, $f(x) = |x|$ .

Practice Problems

Graph the function $f(x) = |x + 3|$ and compare it to the graph of the parent function $f(x) = |x|$ .
Graph the function $f(x) = |x| + 2$ and compare it to the graph of the parent function $f(x) = |x|$ .
Graph the function $f(x) = 2|x|$ and compare it to the graph of the parent function $f(x) = |x|$ .

Answer Key

The graph of $f(x) = |x + 3|$ will be wider than the graph of the parent function $f(x) = |x|$ .
The graph of $f(x) = |x| + 2$ will be wider than the graph of the parent function $f(x) = |x|$ .
The graph of $f(x) = 2|x|$ will be wider than the graph of the parent function $f(x) = |x|$ .

References

[1] Larson, R. (2014). College Algebra. Cengage Learning.
[2] Sullivan, M. (2015). College Algebra. Pearson Education.
[3] Anton, H. (2016). College Algebra. Wiley.
Absolute Value Function Q&A =============================

Q: What is the parent function of absolute value functions?

A: The parent function of absolute value functions is $f(x) = |x|$ . This function is a V-shaped graph with its vertex at the origin (0, 0).

Q: How can absolute value functions be transformed?

A: Absolute value functions can be transformed in various ways, including horizontal shifts, vertical shifts, and vertical stretches or compressions. These transformations can be achieved by adding or subtracting constants from the function, or by multiplying or dividing the function by a constant.

Q: What is a horizontal shift of an absolute value function?

A: A horizontal shift of an absolute value function can be achieved by adding or subtracting a constant from the function. For example, the function $f(x) = |x + 2|$ is a horizontal shift of the parent function $f(x) = |x|$ by 2 units to the left.

Q: What is a vertical shift of an absolute value function?

A: A vertical shift of an absolute value function can be achieved by adding or subtracting a constant from the function. For example, the function $f(x) = |x| - 3$ is a vertical shift of the parent function $f(x) = |x|$ by 3 units down.

Q: What is a vertical stretch or compression of an absolute value function?

A: A vertical stretch or compression of an absolute value function can be achieved by multiplying or dividing the function by a constant. For example, the function $f(x) = 0.5|x|$ is a vertical compression of the parent function $f(x) = |x|$ by a factor of 0.5.

Q: Which function will be narrower than the graph of the parent function, $f(x) = |x|$ ?

A: The function $f(x) = 0.5|x|$ will be narrower than the graph of the parent function, $f(x) = |x|$ . This is because the function is a vertical compression of the parent function by a factor of 0.5, resulting in a narrower graph.

Q: How can I determine which function will be narrower or wider than the graph of the parent function?

A: To determine which function will be narrower or wider than the graph of the parent function, you need to analyze the transformations applied to the function. If the function is shifted horizontally or vertically, it will be wider or narrower than the graph of the parent function. If the function is stretched or compressed vertically, it will be wider or narrower than the graph of the parent function.

Q: What are some common transformations of absolute value functions?

A: Some common transformations of absolute value functions include:

Horizontal shifts: $f(x) = |x + c|$
Vertical shifts: $f(x) = |x| + c$
Vertical stretches: $f(x) = k|x|$
Vertical compressions: $f(x) = \frac{1}{k}|x|$

Q: How can I graph absolute value functions?

A: To graph absolute value functions, you can use the following steps:

Identify the parent function, which is $f(x) = |x|$ .
Determine the transformations applied to the function, such as horizontal or vertical shifts, or vertical stretches or compressions.
Graph the function using the transformations identified in step 2.

Q: What are some common applications of absolute value functions?

A: Some common applications of absolute value functions include:

Modeling distance and speed
Modeling temperature and pressure
Modeling financial transactions and investments
Modeling scientific data and phenomena

Q: How can I use absolute value functions in real-world problems?

A: You can use absolute value functions in real-world problems by applying the transformations and graphing the function. For example, you can use absolute value functions to model the distance between two objects, or to model the temperature of a system.