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)
The function g ( x ) is obtained from f ( x ) by reflecting across the y-axis.
The graph is stretched vertically by a factor of 3.
The graph is shifted 2 units up.
The transformation is reflect across the y -axis, stretch the graph vertically by a factor of 3, shift 2 units up. reflect across the y -axis, stretch vertically by a factor of 3, shift 2 units up
Explanation
Analyze the Functions 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 The term − x in the exponent of g ( x ) indicates a reflection across the y-axis. This is because replacing x with − x in a function reflects the graph across the y-axis.
Vertical Stretch The factor of 3 multiplying the exponential term indicates a vertical stretch by a factor of 3. This is because multiplying a function by a constant greater than 1 stretches the graph vertically.
Vertical Shift The addition of 2 indicates a vertical shift upwards by 2 units. This is because adding a constant to a function shifts the graph vertically upwards.
Conclusion Combining these transformations, we have a reflection across the y-axis, a vertical stretch by a factor of 3, and a vertical shift up 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
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how a wave function transforms can help predict the behavior of particles. In economics, transformations of supply and demand curves can help analyze market changes. In computer graphics, transformations are used to manipulate objects in 2D and 3D space. For instance, if you have a basic sound wave ( f ( x ) = 2 x ) and you want to make it louder (stretch vertically by a factor of 3), reflect it across the y-axis to change its direction ( 2 − x ), and shift it up to adjust its base level (+2), you're applying these transformations in a practical way.
