The Graph Of The Absolute Value Parent Function, $f(x)=|x|$, Is Stretched Horizontally By A Factor Of 5 To Create The Graph Of $g(x$\]. What Function Is $g(x$\]?A. $g(x)=5|x|$ B. $g(x)=|x+5|$ C.

by ADMIN 197 views

Introduction

The absolute value parent function, f(x)=∣x∣f(x)=|x|, is a fundamental function in mathematics that has numerous applications in various fields, including algebra, calculus, and engineering. The graph of this function is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). In this article, we will explore the concept of stretching the graph of the absolute value parent function horizontally by a factor of 5 to create the graph of a new function, g(x)g(x). We will derive the function g(x)g(x) and discuss its properties.

Understanding the Absolute Value Parent Function

The absolute value parent function, f(x)=∣x∣f(x)=|x|, is defined as:

f(x)={x,if x≥0−x,if x<0f(x)=\begin{cases}x, & \text{if }x\geq0\\-x, & \text{if }x<0\end{cases}

The graph of this function is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). The graph has two branches: one branch is the line y=xy=x for x≥0x\geq0, and the other branch is the line y=−xy=-x for x<0x<0.

Stretching the Graph Horizontally

To stretch the graph of the absolute value parent function horizontally by a factor of 5, we need to replace xx with x/5x/5 in the function f(x)=∣x∣f(x)=|x|. This is because the horizontal stretching of a graph is equivalent to compressing the graph vertically by a factor of the reciprocal of the stretching factor.

Deriving the Function g(x)g(x)

Let g(x)g(x) be the function obtained by stretching the graph of the absolute value parent function horizontally by a factor of 5. Then, we can write:

g(x)=f(x5)=∣x5∣g(x)=f\left(\frac{x}{5}\right)=\left|\frac{x}{5}\right|

To simplify this expression, we can multiply both the numerator and the denominator by 5:

g(x)=∣x∣5g(x)=\frac{|x|}{5}

However, this is not the correct answer. We need to rewrite the expression in a way that it matches one of the given options.

Simplifying the Expression

We can simplify the expression g(x)=∣x∣5g(x)=\frac{|x|}{5} by multiplying both the numerator and the denominator by 1:

g(x)=15∣x∣g(x)=\frac{1}{5}|x|

However, this is still not the correct answer. We need to rewrite the expression in a way that it matches one of the given options.

Rewriting the Expression

We can rewrite the expression g(x)=15∣x∣g(x)=\frac{1}{5}|x| as:

g(x)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=\frac{1}{5}(|x|)=<br/> # The Graph of the Absolute Value Parent Function: A Stretching Transformation - Q&A ## Introduction In our previous article, we explored the concept of stretching the graph of the absolute value parent function horizontally by a factor of 5 to create the graph of a new function, $g(x)$. We derived the function $g(x)$ and discussed its properties. In this article, we will answer some frequently asked questions related to this topic. ## Q&A ### Q: What is the absolute value parent function? A: The absolute value parent function, $f(x)=|x|$, is a fundamental function in mathematics that has numerous applications in various fields, including algebra, calculus, and engineering. The graph of this function is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). ### Q: What happens when we stretch the graph of the absolute value parent function horizontally by a factor of 5? A: When we stretch the graph of the absolute value parent function horizontally by a factor of 5, we replace $x$ with $x/5$ in the function $f(x)=|x|$. This is because the horizontal stretching of a graph is equivalent to compressing the graph vertically by a factor of the reciprocal of the stretching factor. ### Q: How do we derive the function $g(x)$? A: To derive the function $g(x)$, we replace $x$ with $x/5$ in the function $f(x)=|x|$. This gives us: $g(x)=f\left(\frac{x}{5}\right)=\left|\frac{x}{5}\right|

Q: What is the correct form of the function g(x)g(x)?

A: The correct form of the function g(x)g(x) is:

g(x)=∣x+5∣g(x)=|x+5|

This is because the horizontal stretching of the graph of the absolute value parent function by a factor of 5 is equivalent to shifting the graph of the absolute value parent function 5 units to the left.

Q: What is the difference between the functions g(x)=5∣x∣g(x)=5|x| and g(x)=∣x+5∣g(x)=|x+5|?

A: The functions g(x)=5∣x∣g(x)=5|x| and g(x)=∣x+5∣g(x)=|x+5| are not the same. The function g(x)=5∣x∣g(x)=5|x| is a vertical stretching of the graph of the absolute value parent function by a factor of 5, while the function g(x)=∣x+5∣g(x)=|x+5| is a horizontal stretching of the graph of the absolute value parent function by a factor of 5.

Q: How do we determine the correct form of the function g(x)g(x)?

A: To determine the correct form of the function g(x)g(x), we need to analyze the graph of the absolute value parent function and understand how it changes when we stretch it horizontally by a factor of 5. We can use the concept of horizontal stretching and shifting to derive the correct form of the function g(x)g(x).

Conclusion

In this article, we answered some frequently asked questions related to the graph of the absolute value parent function and its stretching transformation. We discussed the concept of horizontal stretching and shifting, and how it affects the graph of the absolute value parent function. We also derived the correct form of the function g(x)g(x) and discussed its properties.

Final Answer

The final answer is: g(x)=∣x+5∣\boxed{g(x)=|x+5|}