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 (1)
The function g ( x ) is obtained from f ( x ) by shifting 4 units to the left.
The function is then reflected over the x -axis.
Finally, the function is shifted 2 units down.
Therefore, the transformations are: shift 4 units left, reflect over the x -axis, 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 ) .
Horizontal Shift The transformation x → x + 4 represents a horizontal shift. Since we are replacing x with x + 4 , the graph shifts 4 units to the left.
Reflection The multiplication by − 1 in − ( 2 ) x + 4 represents a reflection. Since the entire function is multiplied by − 1 , the graph is reflected over the x -axis.
Vertical Shift The subtraction of 2 in − ( 2 ) x + 4 − 2 represents a vertical shift. Since we are subtracting 2 from the entire function, the graph shifts 2 units down.
Final Answer Combining these transformations, we have a shift of 4 units to the left, a reflection over the x -axis, and a shift of 2 units down. Therefore, 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 various fields. For example, in physics, understanding how graphs of motion change with different initial conditions helps predict the behavior of objects. In economics, transformations of supply and demand curves can model the effects of taxes or subsidies. In computer graphics, transformations are used to manipulate and animate objects in 3D space. This problem provides a foundation for analyzing and predicting changes in real-world phenomena through mathematical modeling.
