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)
Shift the parent function 4 units to the left: 2 x + 4 .
Reflect the function over the x-axis: − ( 2 ) x + 4 .
Shift the function 2 units down: − ( 2 ) x + 4 − 2 .
The transformations are shift 4 units left, reflect over the x -axis, and shift 2 units down. shift 4 units left, reflect over the x -axis, shift 2 units down
Explanation
Understanding the Problem We are given the parent function f ( x ) = 2 x and the transformed function g ( x ) = − ( 2 ) x + 4 − 2 . We need to describe the transformations that map f ( x ) to g ( x ) .
General Form of Transformations The general form of an exponential transformation is g ( x ) = a ⋅ f ( b ( x − h )) + k , where:
a represents a vertical stretch/compression and reflection over the x-axis if a < 0 .
b represents a horizontal stretch/compression and reflection over the y-axis if b < 0 .
h represents a horizontal shift.
k represents a vertical shift.
Identifying the Transformations In our case, g ( x ) = − ( 2 ) x + 4 − 2 . Comparing this to the parent function f ( x ) = 2 x , we can identify the following transformations:
Horizontal Shift: The term x + 4 in the exponent indicates a horizontal shift. Since it's x + 4 , it means the graph is shifted 4 units to the left.
Reflection: The negative sign in front of the exponential term indicates a reflection over the x-axis.
Vertical Shift: The − 2 at the end of the function indicates a vertical shift of 2 units down.
Combining the Transformations Therefore, the transformations are:
Shift 4 units left
Reflect over the x-axis
Shift 2 units down
Final Answer 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 applied. In economics, it helps in modeling how supply and demand curves shift with changes in market conditions. In computer graphics, transformations are used to manipulate objects in 2D and 3D space. For instance, if you have a basic exponential growth model, f ( x ) = 2 x , and you want to model a scenario where the growth is inverted and shifted due to external factors, you might use a transformation like g ( x ) = − 2 x + 4 − 2 to represent this new scenario.
The transformations from the parent function f ( x ) = 2 x to the function g ( x ) = − ( 2 ) x + 4 − 2 are a shift 4 units left, reflection over the x-axis, and a shift 2 units down. Therefore, the correct answer is option A: shift 4 units left, reflect over the x-axis, shift 2 units down.
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