Which of the following describes the transformations of [tex]g(x)=-(2)^{x+4}-2[/tex] from the parent function [tex]f(x)=2^x[/tex] ?
A. shift 4 units left, reflect over the [tex]x[/tex]-axis, shift 2 units down
B. shift 4 units left, reflect over the [tex]y[/tex]-axis, shift 2 units down
C. shift 4 units right, reflect over the [tex]x[/tex]-axis, shift 2 units down
D. shift 4 units right, reflect over the [tex]y[/tex]-axis, shift 2 units down
Answer (2)
The function g ( x ) = − ( 2 ) x + 4 − 2 is a transformation of the parent function f ( x ) = 2 x .
The term x + 4 in the exponent indicates a horizontal shift of 4 units to the left.
The negative sign in front of the exponential term indicates a reflection over the x-axis.
The − 2 at the end indicates a vertical shift of 2 units down. Therefore, the transformations are shift 4 units left, reflect over the x -axis, shift 2 units down. shift 4 units left, reflect over the x -axis, shift 2 units down
Explanation
Analyze the Problem We are given the parent function f ( x ) = 2 x and the transformed function g ( x ) = − ( 2 ) x + 4 − 2 . Our goal is to describe the transformations that map f ( x ) to g ( x ) . We need to identify the horizontal shift, reflection, and vertical shift.
Identify the Transformations The function g ( x ) can be written as g ( x ) = − 1 ⋅ 2 x + 4 − 2 . Let's analyze each transformation step-by-step:
Horizontal Shift: In the exponent, we have x + 4 . This indicates a horizontal shift. Since it's x + 4 , the graph shifts 4 units to the left .
Reflection: The function is multiplied by − 1 , which means the graph is reflected over the x -axis.
Vertical Shift: Finally, we have a − 2 outside the exponential term. This indicates a vertical shift of 2 units down .
Therefore, the transformations are a shift of 4 units left, a reflection over the x -axis, and a shift of 2 units down.
State the Answer Based on our analysis, the correct answer is:
shift 4 units left, reflect over the x -axis, shift 2 units down
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces is essential. Similarly, in economics, understanding how supply and demand curves shift due to various factors helps in predicting market behavior. In computer graphics, transformations are used to manipulate objects in 2D and 3D space, such as rotating, scaling, and translating them. These transformations are fundamental in creating realistic and interactive visual experiences.
The function g ( x ) = − ( 2 ) x + 4 − 2 undergoes three transformations from the parent function f ( x ) = 2 x : it shifts 4 units left, reflects over the x-axis, and shifts 2 units down. Thus, the answer is A: shift 4 units left, reflect over the x-axis, shift 2 units down.
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