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 f ( x ) = 2 x is reflected across the y-axis due to the − x term in the exponent.
The graph is stretched vertically by a factor of 3 because of the multiplication by 3.
The graph is shifted 2 units up due to the addition of 2.
Therefore, 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 3, shift up by 2
Explanation
Analyze the functions We are given the parent function f ( x ) = 2 x and the transformed function g ( x ) = 3 ( 2 ) − x + 2 . Our goal is to describe the transformations applied to f ( x ) to obtain g ( x ) . We need to identify the effects of the negative sign in the exponent, the multiplication by 3, and the addition of 2.
Reflection The term − x in the exponent of g ( x ) indicates a reflection about the y-axis. This is because replacing x with − x in the function f ( x ) results in f ( − x ) , which is the reflection of f ( x ) across the y-axis.
Vertical Stretch The factor of 3 in g ( x ) indicates a vertical stretch by a factor of 3. Multiplying a function by a constant greater than 1 stretches the graph vertically.
Vertical Shift The addition of 2 in g ( x ) indicates a vertical shift upwards by 2 units. Adding a constant to a function shifts the graph vertically.
Conclusion Combining these transformations, we see that g ( x ) is obtained from f ( x ) by reflecting across the y-axis, stretching vertically by a factor of 3, and shifting 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 useful in many real-world scenarios. For example, in physics, you might use transformations to model the decay of a radioactive substance. The parent function could represent the initial amount of the substance, and transformations could account for the rate of decay and any external factors affecting the process. Similarly, in economics, transformations can be used to model the growth of an investment over time, taking into account factors like interest rates and inflation. By understanding how transformations affect the shape and position of a function's graph, you can gain valuable insights into the underlying phenomena being modeled.
