Which absolute value function, when graphed, will be narrower than the graph of the parent function, [tex]$f(x)=|x|$[/tex]?
A. [tex]$f(x)=|x|-3$[/tex]
B. [tex]$f(x)=|x+2|$[/tex]
C. [tex]$f(x)=0.5|x|$[/tex]
D. [tex]$f(x)=4|x|$[/tex]
Answer (1)
Vertical translations ( f ( x ) = ∣ x ∣ − 3 ) and horizontal translations ( f ( x ) = ∣ x + 2∣ ) do not affect the width of the graph.
Vertical compressions ( f ( x ) = 0.5∣ x ∣ ) make the graph wider.
Vertical stretches ( f ( x ) = 4∣ x ∣ ) make the graph narrower.
Therefore, the function with a narrower graph is f ( x ) = 4∣ x ∣ .
Explanation
Understanding the Problem We are given the parent function f ( x ) = ∣ x ∣ and asked to identify which of the given absolute value functions will have a narrower graph. A narrower graph means the function increases (or decreases) more rapidly than the parent function. This corresponds to a vertical stretch of the parent function.
Analyzing the Options Let's analyze each option:
f ( x ) = ∣ x ∣ − 3 : This is a vertical translation of the parent function. The graph shifts down by 3 units, but the width remains the same.
f ( x ) = ∣ x + 2∣ : This is a horizontal translation of the parent function. The graph shifts left by 2 units, but the width remains the same.
f ( x ) = 0.5∣ x ∣ : This is a vertical compression of the parent function. Since 0.5 < 1 , the graph is wider than the parent function. For example, when x = 2 , f ( 2 ) = 0.5∣2∣ = 1 . For the parent function, when x = 2 , f ( 2 ) = ∣2∣ = 2 . The rate of change is smaller, so the graph is wider.
f ( x ) = 4∣ x ∣ : This is a vertical stretch of the parent function. Since 1"> 4 > 1 , the graph is narrower than the parent function. For example, when x = 1 , f ( 1 ) = 4∣1∣ = 4 . For the parent function, when x = 1 , f ( 1 ) = ∣1∣ = 1 . The rate of change is larger, so the graph is narrower.
Conclusion Therefore, the function f ( x ) = 4∣ x ∣ has a graph that is narrower than the graph of the parent function.
Examples
Imagine you're adjusting the settings on a light dimmer. The parent function, f ( x ) = ∣ x ∣ , represents the standard brightness increase as you turn the knob. If you change the function to f ( x ) = 4∣ x ∣ , you're essentially making the light much more sensitive to each turn, causing the brightness to increase much faster. This is similar to how adjusting the sensitivity of controls in various devices can make them respond more sharply or gradually.
