Which absolute value function, when graphed, represents the parent function, [tex]f(x)=|x|[/tex], reflected over the [tex]x[/tex]-axis and translated 1 unit to the right?
A. [tex]f(x)=-|x|+1[/tex]
B. [tex]f(x)=-|x-1|[/tex]
C. [tex]f(x)=|-x|+1[/tex]
D. [tex]f(x)=|-x-1|[/tex]
Answer (1)
Reflect the parent function f ( x ) = ∣ x ∣ over the x-axis to get f ( x ) = − ∣ x ∣ .
Translate the reflected function 1 unit to the right by replacing x with x − 1 , resulting in f ( x ) = − ∣ x − 1∣ .
The function that represents the parent function reflected over the x-axis and translated 1 unit to the right is f ( x ) = − ∣ x − 1∣ .
Explanation
Understanding the Transformations We are given the parent function f ( x ) = ∣ x ∣ and asked to find the function that represents a reflection over the x-axis and a translation 1 unit to the right.
Reflection over the x-axis Reflecting the parent function f ( x ) = ∣ x ∣ over the x-axis results in the function f ( x ) = − ∣ x ∣ . This is because the y-values are negated.
Translation to the right Translating the function f ( x ) = − ∣ x ∣ one unit to the right means replacing x with ( x − 1 ) . This gives us f ( x ) = − ∣ x − 1∣ .
Final Answer Therefore, the absolute value function that represents the parent function reflected over the x-axis and translated 1 unit to the right is f ( x ) = − ∣ x − 1∣ .
Examples
Imagine you're designing a symmetrical ramp for a skateboard park. The basic shape is an absolute value function, but you need to flip it upside down (reflect over the x-axis) and shift it to the side (translate to the right) to fit the park's layout. Understanding these transformations allows you to adjust the ramp's design precisely to meet the park's specifications. This is also applicable in physics when modeling waveforms or any symmetrical phenomena that needs to be repositioned or inverted.
