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 (2)
Reflect f ( x ) across the y-axis due to the − x term in the exponent.
Stretch the graph vertically by a factor of 3 due to the multiplication by 3.
Shift the graph 2 units up due to the addition of 2.
The transformation is reflect across the y -axis, stretch vertically by 3, and shift up by 2: reflect across the y -axis, stretch vertically by 3, shift up by 2
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 ) .
Rewrite the function The function g ( x ) can be rewritten as g ( x ) = 3 ( 2 − x ) + 2 = 3 ( 2 1 ) x + 2 . Let's analyze the transformations step by step.
Identify the reflection
Reflection: The term − x in the exponent of 2 − x indicates a reflection about the y-axis. This is because replacing x with − x reflects the graph across the y-axis.
Identify the vertical stretch
Vertical Stretch: The factor of 3 multiplying the exponential term, 3 ( 2 ) − x , indicates a vertical stretch by a factor of 3. This means that the y-values of the function are multiplied by 3.
Identify the vertical shift
Vertical Shift: The addition of 2, i.e., 3 ( 2 ) − x + 2 , indicates a vertical shift upwards by 2 units. This means that the entire graph is shifted upwards by 2 units.
Combine the transformations 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.
State the final answer Therefore, the correct description of the transformation of g ( x ) = 3 ( 2 ) − x + 2 from the parent function f ( x ) = 2 x is:
Reflect across the y -axis, stretch the graph vertically by a factor of 3, shift 2 units up.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs transform can help analyze the motion of objects under different conditions. In economics, transformations can model changes in supply and demand curves due to various factors like taxes or subsidies. In computer graphics, transformations are used to manipulate objects in 2D and 3D space, such as scaling, rotating, and translating images or models. These transformations help in creating realistic and interactive visual experiences.
The transformation of the function g ( x ) = 3 ( 2 ) − x + 2 from the parent function f ( x ) = 2 x includes 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 choice is A. reflect across the y-axis, stretch the graph vertically by a factor of 3, shift 2 units up.
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