Question

Which transformations of the graph of (ƒ(x) = 3x) result in the graph of (ƒ(x) = 3 ⋅ 3⁻²ˣ)?

a) Horizontal dilation by 2 and vertical dilation by 3.
b) Horizontal dilation by 0.5, vertical dilation by 3, reflection over the x-axis.
c) Vertical dilation by 2, horizontal dilation by 3, reflection over the x-axis and y-axis.
d) Horizontal dilation by 0.5, vertical dilation by 3, reflection over the y-axis.

Answer 1

To transform the graph of ƒ(x) = 3x into the graph of ƒ(x) = 3 ⋅ 3⁻²ˣ, you need to apply a horizontal dilation by 2 and a vertical dilation by 3.

Step-by-step explanation:

To transform the graph of ƒ(x) = 3x into the graph of ƒ(x) = 3 ⋅ 3⁻²ˣ, we need to apply horizontal and vertical dilations and possibly reflections.

1. A horizontal dilation by 2 stretches the graph horizontally, while a vertical dilation by 3 stretches the graph vertically. This option matches the given transformation.
2. This option includes a horizontal dilation by 0.5, a vertical dilation by 3, and a reflection over the x-axis. The reflection over the x-axis changes the slope from positive to negative, so it does not match the given transformation.
3. This option includes a vertical dilation by 2, a horizontal dilation by 3, and reflections over the x-axis and y-axis. The reflections change the slopes of the graph and do not match the given transformation.
4. This option includes a horizontal dilation by 0.5, a vertical dilation by 3, and a reflection over the y-axis. The reflection over the y-axis changes the slope from positive to negative, so it does not match the given transformation.

Therefore, the correct option is a) Horizontal dilation by 2 and vertical dilation by 3.