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 to Absolute Value Functions

Absolute value functions are a fundamental concept in mathematics, particularly in algebra and calculus. These functions have a unique graph that is shaped like a "V" and are used to model various real-world phenomena. In this article, we will delve into the world of absolute value functions and explore the characteristics of their 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 symmetrical about the y-axis and has a minimum value of 0. The graph of $f(x) = |x|$ is a "V" shape, with the vertex at the origin (0, 0).

Understanding the Graph of $f(x) = |x|$

To understand the graph of $f(x) = |x|$, let's consider the following:

• When $x$ is positive, the graph of $f(x) = |x|$ is a straight line with a slope of 1.
• When $x$ is negative, the graph of $f(x) = |x|$ is a straight line with a slope of -1.
• The graph of $f(x) = |x|$ is symmetrical about the y-axis, meaning that for every point $(x, y)$ on the graph, there is a corresponding point $(-x, y)$.

Translating the Parent Function

When we translate the parent function $f(x) = |x|$ to the right by 2 units, we get a new function $f(x) = |x - 2|$. This function has a graph that is identical to the graph of $f(x) = |x|$, but shifted 2 units to the right.

Comparing the Graphs

Now, let's compare the graphs of the given functions:

• $f(x) = 1.3|x| - 2$
• $f(x) = 3|x - 2|$
• $f(x) = \frac{3}{4}|x - 2|$

Analyzing Option A: $f(x) = 1.3|x| - 2$

Option A is a function that is a vertical stretch of the parent function $f(x) = |x|$ by a factor of 1.3, followed by a vertical shift down by 2 units. However, this function is not translated to the right by 2 units, so it does not meet the criteria.

Analyzing Option B: $f(x) = 3|x - 2|$

Option B is a function that is a vertical stretch of the parent function $f(x) = |x|$ by a factor of 3, followed by a horizontal shift 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, making it a strong candidate.

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

Option C is a function that is a vertical compression of the parent function $f(x) = |x|$ by a factor of $\frac{3}{4}$, followed by a horizontal shift to the right by 2 units. This function has a graph that is narrower than the parent function, so it does not meet the criteria.

Conclusion

In conclusion, the function that has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units is $f(x) = 3|x - 2|$. This function meets the criteria and has a graph that is identical to the graph of the parent function, but shifted 2 units to the right.

Final Answer

The final answer is:

• B. $f(x) = 3|x - 2|$

This function has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units.

Introduction to Absolute Value Functions

Absolute value functions are a fundamental concept in mathematics, particularly in algebra and calculus. These functions have a unique graph that is shaped like a "V" and are used to model various real-world phenomena. In this article, we will delve into the world of absolute value functions and explore the characteristics of their 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 symmetrical about the y-axis and has a minimum value of 0. The graph of $f(x) = |x|$ is a "V" shape, with the vertex at the origin (0, 0).

Understanding the Graph of $f(x) = |x|$

To understand the graph of $f(x) = |x|$, let's consider the following:

• When $x$ is positive, the graph of $f(x) = |x|$ is a straight line with a slope of 1.
• When $x$ is negative, the graph of $f(x) = |x|$ is a straight line with a slope of -1.
• The graph of $f(x) = |x|$ is symmetrical about the y-axis, meaning that for every point $(x, y)$ on the graph, there is a corresponding point $(-x, y)$.

Translating the Parent Function

When we translate the parent function $f(x) = |x|$ to the right by 2 units, we get a new function $f(x) = |x - 2|$. This function has a graph that is identical to the graph of $f(x) = |x|$, but shifted 2 units to the right.

Comparing the Graphs

Now, let's compare the graphs of the given functions:

• $f(x) = 1.3|x| - 2$
• $f(x) = 3|x - 2|$
• $f(x) = \frac{3}{4}|x - 2|$

Analyzing Option A: $f(x) = 1.3|x| - 2$

Option A is a function that is a vertical stretch of the parent function $f(x) = |x|$ by a factor of 1.3, followed by a vertical shift down by 2 units. However, this function is not translated to the right by 2 units, so it does not meet the criteria.

Analyzing Option B: $f(x) = 3|x - 2|$

Option B is a function that is a vertical stretch of the parent function $f(x) = |x|$ by a factor of 3, followed by a horizontal shift 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, making it a strong candidate.

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

Option C is a function that is a vertical compression of the parent function $f(x) = |x|$ by a factor of $\frac{3}{4}$, followed by a horizontal shift to the right by 2 units. This function has a graph that is narrower than the parent function, so it does not meet the criteria.

Conclusion

In conclusion, the function that has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units is $f(x) = 3|x - 2|$. This function meets the criteria and has a graph that is identical to the graph of the parent function, but shifted 2 units to the right.

Final Answer

The final answer is:

• B. $f(x) = 3|x - 2|$

This function has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units.

Q&A: Absolute Value Functions

Q1: What is the parent function of absolute value functions?

A1: The parent function of absolute value functions is $f(x) = |x|$.

Q2: What is the graph of $f(x) = |x|$?

A2: The graph of $f(x) = |x|$ is a "V" shape, with the vertex at the origin (0, 0).

Q3: What happens when we translate the parent function $f(x) = |x|$ to the right by 2 units?

A3: When we translate the parent function $f(x) = |x|$ to the right by 2 units, we get a new function $f(x) = |x - 2|$.

Q4: Which function has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units?

A4: The function that has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units is $f(x) = 3|x - 2|$.

Q5: What is the final answer?

A5: The final answer is:

• B. $f(x) = 3|x - 2|$

This function has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units.

Q6: What is the purpose of absolute value functions?

A6: Absolute value functions are used to model various real-world phenomena, such as the distance between two points, the absolute value of a quantity, and the magnitude of a vector.

Q7: How do we determine the graph of an absolute value function?

A7: To determine the graph of an absolute value function, we need to consider the following:

• When $x$ is positive, the graph of $f(x) = |x|$ is a straight line with a slope of 1.
• When $x$ is negative, the graph of $f(x) = |x|$ is a straight line with a slope of -1.
• The graph of $f(x) = |x|$ is symmetrical about the y-axis, meaning that for every point $(x, y)$ on the graph, there is a corresponding point $(-x, y)$.

Q8: What is the difference between a vertical stretch and a vertical compression of an absolute value function?

A8: A vertical stretch of an absolute value function is a transformation that makes the graph wider, while a vertical compression of an absolute value function is a transformation that makes the graph narrower.

Q9: How do we translate an absolute value function to the right by 2 units?

A9: To translate an absolute value function to the right by 2 units, we need to replace $x$ with $x - 2$ in the function.

Q10: What is the final answer to the problem?

A10: The final answer to the problem is:

• B. $f(x) = 3|x - 2|$

This function has a graph that is wider than the parent function $f(x) = |x|$ and is translated to the right by 2 units.