Let's start by analyzing the transformations step by step based on the given function $f(x) = 4^x$.

(a) Shifting $f(x) = 4^x$ 9 units downward:

• To shift the graph downward by 9 units, we subtract 9 from the original function.
• The new equation becomes: $y = 4^x - 9$

(b) Shifting $f(x) = 4^x$ 7 units to the right:

• To shift the graph to the right by 7 units, we replace $x$ with $x - 7$.
• The new equation becomes: $y = 4^{x - 7}$

(c) Reflecting $f(x) = 4^x$ about the y-axis:

• To reflect the graph about the y-axis, we replace $x$ with $-x$.
• The new equation becomes: $y = 4^{-x}$

Final Equations:

• (a) Shifting 9 units downward: $y = 4^x - 9$
• (b) Shifting 7 units to the right: $y = 4^{x - 7}$
• (c) Reflecting about the y-axis: $y = 4^{-x}$

Do you want more details on any of these transformations, or have any questions?

Here are some related questions to explore:

1. What happens when the graph is shifted 5 units upward?
2. How would the equation change if the graph were shifted left instead of right?
3. What is the impact of reflecting $f(x)$ about the x-axis instead of the y-axis?
4. How does changing the base of the exponential function (e.g., from 4 to 2) affect the graph?
5. Can you combine multiple transformations in a single equation?

Tip: When reflecting about the y-axis, the graph's growth behavior switches direction, but the rate remains the same.