Which Absolute Value Function, When Graphed, Will Be Narrower Than The Graph Of The Parent Function, F ( X ) = ∣ X ∣ F(x) = |x| F ( X ) = ∣ X ∣ ?A. F ( X ) = ∣ X ∣ − 3 F(x) = |x| - 3 F ( X ) = ∣ X ∣ − 3 B. F ( X ) = ∣ X + 2 ∣ F(x) = |x + 2| F ( X ) = ∣ X + 2∣ C. F ( X ) = 0.5 ∣ X ∣ F(x) = 0.5|x| F ( X ) = 0.5∣ X ∣ D. F ( X ) = 4 ∣ X ∣ F(x) = 4|x| F ( X ) = 4∣ X ∣

by ADMIN 366 views

Introduction

Absolute value functions are a fundamental concept in mathematics, and their graphs play a crucial role in various mathematical and real-world applications. The parent function, f(x)=∣x∣f(x) = |x|, is a basic absolute value function that represents the distance of a point from the origin on the number line. In this article, we will explore which absolute value function, when graphed, will be narrower than the graph of the parent function.

Parent Function: f(x)=∣x∣f(x) = |x|

The parent function, f(x)=∣x∣f(x) = |x|, is a V-shaped graph that opens upwards and has its vertex at the origin (0, 0). The graph of the parent function is symmetric about the y-axis and has a slope of 1 on the right side and a slope of -1 on the left side.

Graphing Absolute Value Functions

To graph an absolute value function, we need to consider the following cases:

  • If x≥0x \geq 0, then f(x)=xf(x) = x.
  • If x<0x < 0, then f(x)=−xf(x) = -x.

Using these cases, we can graph the absolute value function by plotting the points (x,x)(x, x) for x≥0x \geq 0 and (−x,x)(-x, x) for x<0x < 0.

Comparing Graphs

To determine which absolute value function will be narrower than the graph of the parent function, we need to compare the graphs of the given functions.

Option A: f(x)=∣x∣−3f(x) = |x| - 3

The graph of f(x)=∣x∣−3f(x) = |x| - 3 is a V-shaped graph that opens upwards and has its vertex at the point (0, -3). The graph is shifted downwards by 3 units compared to the parent function.

**Graph of $f(x) = |x| - 3$**
-----------------------------

The graph of f(x)=∣x∣−3f(x) = |x| - 3 is narrower than the graph of the parent function because it is shifted downwards by 3 units. However, the graph is still the same shape as the parent function, and the width of the graph remains the same.


Option B: f(x)=∣x+2∣f(x) = |x + 2|


The graph of f(x)=∣x+2∣f(x) = |x + 2| is a V-shaped graph that opens upwards and has its vertex at the point (-2, 0). The graph is shifted to the left by 2 units compared to the parent function.


**Graph of $f(x) = |x + 2|$
-----------------------------

The graph of $f(x) = |x + 2|$ is narrower than the graph of the parent function because it is shifted to the left by 2 units. However, the graph is still the same shape as the parent function, and the width of the graph remains the same.

### Option C: $f(x) = 0.5|x|$

The graph of $f(x) = 0.5|x|$ is a V-shaped graph that opens upwards and has its vertex at the origin (0, 0). The graph is compressed vertically by a factor of 0.5 compared to the parent function.

```markdown
**Graph of $f(x) = 0.5|x|$
-----------------------------

The graph of $f(x) = 0.5|x|$ is narrower than the graph of the parent function because it is compressed vertically by a factor of 0.5. The width of the graph remains the same, but the height of the graph is reduced.

### Option D: $f(x) = 4|x|$

The graph of $f(x) = 4|x|$ is a V-shaped graph that opens upwards and has its vertex at the origin (0, 0). The graph is stretched vertically by a factor of 4 compared to the parent function.

```markdown
**Graph of $f(x) = 4|x|$
-----------------------------

The graph of $f(x) = 4|x|$ is wider than the graph of the parent function because it is stretched vertically by a factor of 4. The width of the graph remains the same, but the height of the graph is increased.

**Conclusion**
--------------

Based on the analysis of the graphs of the given absolute value functions, we can conclude that the graph of $f(x) = 0.5|x|$ is the only option that is narrower than the graph of the parent function. The graph of $f(x) = 0.5|x|$ is compressed vertically by a factor of 0.5, resulting in a narrower graph compared to the parent function.

**Key Takeaways**
------------------

*   The parent function, $f(x) = |x|$, is a basic absolute value function that represents the distance of a point from the origin on the number line.
*   To graph an absolute value function, we need to consider the cases $x \geq 0$ and $x &lt; 0$.
*   The graph of $f(x) = 0.5|x|$ is narrower than the graph of the parent function because it is compressed vertically by a factor of 0.5.
*   The graph of $f(x) = |x| - 3$ is shifted downwards by 3 units, but the width of the graph remains the same.
*   The graph of $f(x) = |x + 2|$ is shifted to the left by 2 units, but the width of the graph remains the same.
*   The graph of $f(x) = 4|x|$ is stretched vertically by a factor of 4, resulting in a wider graph compared to the parent function.&lt;br/&gt;
**Absolute Value Functions: Q&amp;A**
=============================

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

A: The parent function of absolute value functions is $f(x) = |x|$, which represents the distance of a point from the origin on the number line.

**Q: How do you graph an absolute value function?**
----------------------------------------------

A: To graph an absolute value function, you need to consider the cases $x \geq 0$ and $x &lt; 0$. For $x \geq 0$, the graph is a line segment from $(0, 0)$ to $(x, x)$. For $x &lt; 0$, the graph is a line segment from $(0, 0)$ to $(-x, x)$.

**Q: What is the difference between the graph of $f(x) = |x|$ and $f(x) = |x + 2|$?**
--------------------------------------------------------------------------------

A: The graph of $f(x) = |x + 2|$ is shifted to the left by 2 units compared to the graph of $f(x) = |x|$. This means that the vertex of the graph of $f(x) = |x + 2|$ is at the point $(-2, 0)$, whereas the vertex of the graph of $f(x) = |x|$ is at the point $(0, 0)$.

**Q: How does the graph of $f(x) = 0.5|x|$ compare to the graph of $f(x) = |x|$?**
--------------------------------------------------------------------------------

A: The graph of $f(x) = 0.5|x|$ is compressed vertically by a factor of 0.5 compared to the graph of $f(x) = |x|$. This means that the height of the graph of $f(x) = 0.5|x|$ is reduced by a factor of 0.5, resulting in a narrower graph.

**Q: What is the effect of multiplying the absolute value function by a constant?**
--------------------------------------------------------------------------------

A: Multiplying the absolute value function by a constant has the effect of stretching or compressing the graph vertically. If the constant is greater than 1, the graph is stretched vertically. If the constant is less than 1, the graph is compressed vertically.

**Q: Can you give an example of an absolute value function that is wider than the graph of $f(x) = |x|$?**
-----------------------------------------------------------------------------------------

A: Yes, an example of an absolute value function that is wider than the graph of $f(x) = |x|$ is $f(x) = 4|x|$. The graph of $f(x) = 4|x|$ is stretched vertically by a factor of 4 compared to the graph of $f(x) = |x|$, resulting in a wider graph.

**Q: How do you determine which absolute value function is narrower than the graph of $f(x) = |x|$?**
-----------------------------------------------------------------------------------------

A: To determine which absolute value function is narrower than the graph of $f(x) = |x|$, you need to compare the graphs of the given functions. If the graph of the absolute value function is compressed vertically by a factor of less than 1, it is narrower than the graph of $f(x) = |x|$.

**Q: What is the significance of absolute value functions in real-world applications?**
--------------------------------------------------------------------------------

A: Absolute value functions have numerous applications in real-world problems, such as physics, engineering, and economics. They are used to model problems involving distance, speed, and time, and to solve equations involving absolute values.

**Q: Can you give an example of a real-world application of absolute value functions?**
--------------------------------------------------------------------------------

A: Yes, an example of a real-world application of absolute value functions is the calculation of distance and speed in physics. The absolute value function is used to calculate the distance traveled by an object, and the speed of the object, given its initial and final positions and times.

**Conclusion**
--------------

Absolute value functions are a fundamental concept in mathematics, and their graphs play a crucial role in various mathematical and real-world applications. By understanding the properties and behavior of absolute value functions, we can solve equations and model problems involving distance, speed, and time.</code></pre>