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 (1)
The function g ( x ) is obtained from f ( x ) by shifting 4 units to the left.
The function is then reflected over the x-axis.
Finally, the function is shifted 2 units down.
Therefore, the transformations are shift 4 units left, reflect over the x -axis, shift 2 units down, and the answer is shift 4 units left, reflect over the x -axis, shift 2 units down .
Explanation
Analyze the Transformations 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 any horizontal shifts, reflections, and vertical shifts.
Identify the Transformations Let's break down the transformations step by step:
Horizontal Shift: The term x + 4 in the exponent indicates a horizontal shift. Since it's x + 4 , the graph shifts 4 units to the left .
Reflection: The negative sign in front of the exponential term, − ( 2 ) x + 4 , indicates a reflection. Since the negative sign is outside the function, it's a reflection over the x-axis .
Vertical Shift: The term − 2 at the end of the function, − ( 2 ) x + 4 − 2 , indicates a vertical shift. Since it's − 2 , the graph shifts 2 units down .
Combine the Transformations Combining these transformations, we have:
Shift 4 units left
Reflect over the x-axis
Shift 2 units down
This corresponds to the first option.
State the Final Answer Therefore, the transformations of g ( x ) = − ( 2 ) x + 4 − 2 from the parent function f ( x ) = 2 x are a shift 4 units left, a reflection over the x -axis, and a 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 involves transformations. In economics, shifts in supply and demand curves can be described using transformations of functions. In computer graphics, transformations are used to manipulate objects in 2D and 3D space, such as scaling, rotating, and translating them. These transformations are essential for creating realistic and interactive visual experiences.
