Which Absolute Value Function Has A Graph That Is Wider Than The Parent Function, F ( X ) = ∣ X ∣ F(x)=|x| F ( X ) = ∣ X ∣ , And Is Translated To The Right 2 Units?A. F ( X ) = 1.3 ∣ X ∣ − 2 F(x)=1.3|x|-2 F ( X ) = 1.3∣ X ∣ − 2 B. F ( X ) = 3 ∣ X − 2 ∣ F(x)=3|x-2| F ( X ) = 3∣ X − 2∣ C. F ( X ) = 3 4 ∣ X − 2 ∣ F(x)=\frac{3}{4}|x-2| F ( X ) = 4 3 ​ ∣ X − 2∣ D.

Introduction

Absolute value functions are a fundamental concept in mathematics, particularly in algebra and calculus. These functions have a unique graph that is V-shaped, with the vertex at the origin (0, 0). In this article, we will explore the properties of absolute value functions and how they can be translated to create new functions with different graphs.

The Parent Function: $f(x) = |x|$

The parent function of absolute value functions is $f(x) = |x|$. This function has a graph that is V-shaped, with the vertex at the origin (0, 0). The graph is symmetric about the y-axis, and the absolute value of the function is always non-negative.

Graph Properties

The graph of $f(x) = |x|$ has several important properties:

• The graph is V-shaped, with the vertex at the origin (0, 0).
• The graph is symmetric about the y-axis.
• The absolute value of the function is always non-negative.
• The graph has a minimum value of 0 at the origin (0, 0).

Translations of Absolute Value Functions

Translations of absolute value functions involve shifting the graph of the parent function to the left or right by a certain number of units. In this article, we will focus on translations to the right.

Translation to the Right: $f(x) = |x - h|$

When the graph of the parent function is translated to the right by $h$ units, the new function is given by $f(x) = |x - h|$. This function has a graph that is also V-shaped, but the vertex is now located at the point $(h, 0)$.

Example: $f(x) = |x - 2|$

Let's consider an example of a translation to the right by 2 units. The new function is given by $f(x) = |x - 2|$. This function has a graph that is V-shaped, with the vertex at the point $(2, 0)$.

Scaling the Graph: $f(x) = a|x - h|$

When the graph of the parent function is scaled by a factor of $a$, the new function is given by $f(x) = a|x - h|$. This function has a graph that is also V-shaped, but the height of the graph is now scaled by a factor of $a$.

Example: $f(x) = 3|x - 2|$

Let's consider an example of a scaling by a factor of 3. The new function is given by $f(x) = 3|x - 2|$. This function has a graph that is V-shaped, with the vertex at the point $(2, 0)$, and the height of the graph is scaled by a factor of 3.

Which Absolute Value Function Has a Graph That is Wider Than the Parent Function and is Translated to the Right 2 Units?

Now that we have discussed the properties of absolute value functions and how they can be translated, let's consider the options given in the problem.

A. $f(x) = 1.3|x| - 2$

This function is a scaling of the parent function by a factor of 1.3, but it is not translated to the right by 2 units.

B. $f(x) = 3|x - 2|$

This function is a scaling of the parent function by a factor of 3 and is translated to the right by 2 units. This function has a graph that is wider than the parent function and is translated to the right by 2 units.

C. $f(x) = \frac{3}{4}|x - 2|$

This function is a scaling of the parent function by a factor of $\frac{3}{4}$ and is translated to the right by 2 units. This function has a graph that is narrower than the parent function and is translated to the right by 2 units.

D. $f(x) = |x - 2|$

This function is the parent function translated to the right by 2 units, but it is not scaled by a factor greater than 1.

Conclusion

In conclusion, the absolute value function that has a graph that is wider than the parent function and is translated to the right by 2 units is $f(x) = 3|x - 2|$. This function has a graph that is V-shaped, with the vertex at the point $(2, 0)$, and the height of the graph is scaled by a factor of 3.

References

• [1] "Absolute Value Functions." Math Open Reference, mathopenref.com/absolutevalue.html.
• [2] "Graphing Absolute Value Functions." Purplemath, purplemath.com/modules/graphs/absvalue.htm.
• [3] "Translations of Absolute Value Functions." Khan Academy, khanacademy.org/math/algebra/absolute-value/translations-of-absolute-value-functions/v/translations-of-absolute-value-functions.
  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 has a graph that is V-shaped, with the vertex at the origin (0, 0).

Q: What are the properties of the graph of $f(x) = |x|$?

A: The graph of $f(x) = |x|$ has several important properties:

• The graph is V-shaped, with the vertex at the origin (0, 0).
• The graph is symmetric about the y-axis.
• The absolute value of the function is always non-negative.
• The graph has a minimum value of 0 at the origin (0, 0).

Q: What is the effect of translating the graph of $f(x) = |x|$ to the right by $h$ units?

A: When the graph of the parent function is translated to the right by $h$ units, the new function is given by $f(x) = |x - h|$. This function has a graph that is also V-shaped, but the vertex is now located at the point $(h, 0)$.

Q: What is the effect of scaling the graph of $f(x) = |x|$ by a factor of $a$?

A: When the graph of the parent function is scaled by a factor of $a$, the new function is given by $f(x) = a|x - h|$. This function has a graph that is also V-shaped, but the height of the graph is now scaled by a factor of $a$.

Q: Which absolute value function has a graph that is wider than the parent function and is translated to the right by 2 units?

A: The absolute value function that has a graph that is wider than the parent function and is translated to the right by 2 units is $f(x) = 3|x - 2|$.

Q: What is the difference between $f(x) = 3|x - 2|$ and $f(x) = |x - 2|$?

A: The function $f(x) = 3|x - 2|$ is a scaling of the parent function by a factor of 3 and is translated to the right by 2 units. The function $f(x) = |x - 2|$ is the parent function translated to the right by 2 units, but it is not scaled by a factor greater than 1.

Q: How do you determine the width of the graph of an absolute value function?

A: The width of the graph of an absolute value function can be determined by the factor by which the parent function is scaled. If the factor is greater than 1, the graph is wider than the parent function. If the factor is less than 1, the graph is narrower than the parent function.

Q: Can you give an example of an absolute value function that is translated to the left by 2 units?

A: Yes, an example of an absolute value function that is translated to the left by 2 units is $f(x) = |x + 2|$.

Q: Can you give an example of an absolute value function that is scaled by a factor of $\frac{1}{2}$?

A: Yes, an example of an absolute value function that is scaled by a factor of $\frac{1}{2}$ is $f(x) = \frac{1}{2}|x|$.

Q: What is the vertex of the graph of $f(x) = |x - 2|$?

A: The vertex of the graph of $f(x) = |x - 2|$ is located at the point $(2, 0)$.

Q: What is the minimum value of the graph of $f(x) = |x - 2|$?

A: The minimum value of the graph of $f(x) = |x - 2|$ is 0, which occurs at the vertex $(2, 0)$.

Q: Is the graph of $f(x) = |x - 2|$ symmetric about the y-axis?

A: Yes, the graph of $f(x) = |x - 2|$ is symmetric about the y-axis.