To determine the correct statement about how the function [tex]\( h(x) = -3 \cdot 2^x \)[/tex] is a transformation of the parent function [tex]\( f(x) = 2^x \)[/tex], let's analyze the components of [tex]\( h(x) \)[/tex] step by step.
1. Reflection:
- The parent function [tex]\( f(x) = 2^x \)[/tex] is modified by the coefficient [tex]\(-3\)[/tex]. The negative sign in front of the coefficient indicates a reflection across the x-axis. Multiplying [tex]\( f(x) \)[/tex] by [tex]\(-1\)[/tex] will flip it upside-down.
2. Vertical Stretch/Dilation:
- The coefficient [tex]\(3\)[/tex] (the absolute value of [tex]\(-3\)[/tex]) indicates a vertical stretch, or dilation, by a factor of 3. This means that all values of [tex]\(2^x\)[/tex] are scaled by a factor of 3. Thus, each point on the graph of [tex]\(2^x\)[/tex] will be three times further away from the x-axis than it was before the transformation.
Combining these transformations, [tex]\( h(x) = -3 \cdot 2^x \)[/tex] both reflects [tex]\( f(x) \)[/tex] across the x-axis and scales it vertically by a factor of 3.
Therefore, the correct statement is:
A. Function [tex]\( h \)[/tex] is a reflection and a dilation of function [tex]\( f \)[/tex].
Thus, the correct answer is:
A. Function [tex]\( h \)[/tex] is a reflection and a dilation of function [tex]\( f \)[/tex].
