Which of the following describes the transformation of $g(x)=3(2)^{-x}+2$ from the parent function $f(x)=2^x$?
A. reflect across the $x$-axis, stretch the graph vertically by a factor of 3, shift 2 units up
B. reflect across the $y$-axis, stretch the graph vertically by a factor of 2, shift 3 units up
C. reflect across the $x$-axis, stretch the graph vertically by a factor of 2, shift 3 units up
D. reflect across the $y$-axis, stretch the graph vertically by a factor of 3, shift 2 units up
Answer (1)
Reflect across the y -axis: 2 x becomes 2 − x .
Stretch the graph vertically by a factor of 3: 2 − x becomes 3 ( 2 ) − x .
Shift the graph 2 units up: 3 ( 2 ) − x becomes 3 ( 2 ) − x + 2 .
The transformation is reflect across the y -axis, stretch the graph vertically by a factor of 3, shift 2 units up.
Explanation
Analyze the problem We are given the parent function f ( x ) = 2 x and the transformed function g ( x ) = 3 ( 2 ) − x + 2 . We need to describe the transformations applied to f ( x ) to obtain g ( x ) .
Reflection First, let's analyze the exponent. The term − x in 2 − x indicates a reflection across the y-axis. So, 2 − x is the reflection of 2 x across the y-axis.
Vertical Stretch Next, we observe the factor of 3 multiplying the exponential term, 3 ( 2 ) − x . This indicates a vertical stretch by a factor of 3.
Vertical Shift Finally, the addition of 2, 3 ( 2 ) − x + 2 , indicates a vertical shift upwards by 2 units.
Conclusion Combining these transformations, we have a reflection across the y-axis, a vertical stretch by a factor of 3, and a vertical shift upwards by 2 units. Therefore, the correct answer is: reflect across the y -axis, stretch the graph vertically by a factor of 3, shift 2 units up
Examples
Imagine you are adjusting the settings on an audio equalizer. The parent function could represent the original sound wave. Reflecting across the y-axis might reverse the audio, stretching vertically amplifies certain frequencies, and shifting upwards increases the overall volume. Understanding these transformations allows you to fine-tune the sound to your preference. This concept is also used in image processing, where transformations like scaling, rotation, and translation are applied to manipulate images.
