Which absolute value function, when graphed, will be wider than the graph of the parent function, [tex]f(x)=|x|[/tex]?
A. [tex]f(x)=|x|+3[/tex]
B. [tex]f(x)=|x-6|[/tex]
C. [tex]f(x)=\frac{1}{3}|x|[/tex]
D. [tex]f(x)=9|x|[/tex]
Answer (1)
Vertical compressions (multiplying by a factor less than 1) make the graph wider.
Vertical stretches (multiplying by a factor greater than 1) make the graph narrower.
Translations (adding or subtracting constants inside or outside the absolute value) do not affect the width.
Therefore, f ( x ) = 3 1 ∣ x ∣ is wider than f ( x ) = ∣ x ∣ . f ( x ) = 3 1 ∣ x ∣
Explanation
Understanding the Problem We are given the parent function f ( x ) = ∣ x ∣ and four other absolute value functions. We need to determine which of the given functions has a graph that is wider than the graph of the parent function. A wider graph corresponds to a horizontal stretch or a vertical compression.
Analyzing the General Form The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where:
a affects the vertical stretch or compression. If ∣ a ∣ < 1 , the graph is wider (vertical compression). If 1"> ∣ a ∣ > 1 , the graph is narrower (vertical stretch).
h affects the horizontal translation. A change in h shifts the graph left or right but does not change the width.
k affects the vertical translation. A change in k shifts the graph up or down but does not change the width.
Analyzing Each Function Let's analyze each of the given functions:
f ( x ) = ∣ x ∣ + 3 : This is a vertical translation ( k = 3 ). The width of the graph is the same as the parent function.
f ( x ) = ∣ x − 6∣ : This is a horizontal translation ( h = 6 ). The width of the graph is the same as the parent function.
f ( x ) = 3 1 ∣ x ∣ : Here, a = 3 1 . Since ∣ 3 1 ∣ < 1 , the graph is wider than the parent function.
f ( x ) = 9∣ x ∣ : Here, a = 9 . Since 1"> ∣9∣ > 1 , the graph is narrower than the parent function.
Conclusion Therefore, the function that will have a wider graph than the parent function is f ( x ) = 3 1 ∣ x ∣ .
Examples
Imagine you're stretching or compressing a rubber band. The parent function |x| is like the original rubber band. When you multiply the absolute value by a fraction less than 1, like 1/3, it's like compressing the rubber band vertically, making it wider. This concept is used in image editing to stretch or compress images, or in economics to model how changes in one variable affect another.
