Which absolute value function has a graph that is wider than the parent function, [tex]f(x)=|x|[/tex], and is translated to the right 2 units?
A. [tex]f(x)=1.3|x|-2[/tex]
B. [tex]f(x)=3|x-2|[/tex]
C. [tex]f(x)=\frac{3}{4}|x-2|[/tex]
D. [tex]f(x)=\frac{4}{3}|x|+2[/tex]
Answer (1)
A horizontal translation to the right by 2 units is achieved by replacing x with ( x − 2 ) .
A vertical compression (making the graph wider) is achieved by multiplying the absolute value by a constant between 0 and 1.
The function f ( x ) = 4 3 ∣ x − 2∣ is wider than the parent function and is translated to the right 2 units.
Therefore, the answer is f ( x ) = 4 3 ∣ x − 2∣ .
Explanation
Understanding the Problem We are looking for an absolute value function that is wider than the parent function, f ( x ) = ∣ x ∣ , and is translated to the right 2 units. Let's analyze what each transformation does to the parent function.
Key Transformations A horizontal translation to the right by 2 units is achieved by replacing x with ( x − 2 ) in the parent function. A vertical compression (making the graph wider) is achieved by multiplying the absolute value by a constant between 0 and 1.
Analyzing Each Option Let's examine each option:
f ( x ) = 1.3∣ x ∣ − 2 : This is a vertical stretch by a factor of 1.3 (making it narrower) and a vertical translation down by 2 units. It is not wider and not translated to the right.
f ( x ) = 3∣ x − 2∣ : This is a vertical stretch by a factor of 3 (making it narrower) and a horizontal translation to the right by 2 units. It is not wider.
f ( x ) = 4 3 ∣ x − 2∣ : This is a vertical compression by a factor of 4 3 (making it wider) and a horizontal translation to the right by 2 units. This satisfies both conditions.
f ( x ) = 3 4 ∣ x ∣ + 2 : This is a vertical stretch by a factor of 3 4 (making it narrower) and a vertical translation up by 2 units. It is not wider and not translated to the right.
Conclusion Therefore, the correct option is f ( x ) = 4 3 ∣ x − 2∣ .
Examples
Imagine you're adjusting the settings on a camera. The parent function |x| represents the basic view. Making the graph wider (vertical compression) is like zooming out to capture more of the scene. Translating the graph to the right is like shifting the camera's focus to a specific point. In this case, you're zooming out by a factor of 3/4 and shifting the focus 2 units to the right. This is useful in photography, image processing, and even in adjusting the display on your phone or computer screen.
