Question

Which transformations of the graph of f(x) = 3ˣ result in the graph of f(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

Horizontal dilation by 0.5, vertical dilation by 3, reflection over the x-axis transformations of the graph of f(x) = 3ˣ result in the graph of f(x) = 3 ⋅ 3⁻²ˣ Therefore, correct answer is b) Horizontal dilation by 0.5, vertical dilation by 3, reflection over the x-axis

Step-by-step explanation:

To transform the graph of
`\(f(x) = 3^x\) into \(f(x) = 3 \cdot 3^(-2x)\),` we need to apply a horizontal dilation by 0.5, a vertical dilation by 3, and a reflection over the x-axis. The horizontal dilation by 0.5 is represented by the exponent -2x, the vertical dilation by 3 is due to the coefficient 3 in front of the exponential term, and the reflection over the x-axis is indicated by the negation of the exponent in
`\(3^(-2x)\).`

Understanding how different transformations affect the graphs of exponential functions is crucial in mathematics. These transformations, including dilations and reflections, play a significant role in manipulating functions to fit specific requirements or to model various real-world scenarios.

Therefore, correct answer is b) Horizontal dilation by 0.5, vertical dilation by 3, reflection over the x-axis