Let's go through each transformation one by one:

Given function:

The original function is $f(x) = 8^x$.

Part (a): Shifting $f(x)$ 3 units downward

To shift the function downward, we subtract 3 from $f(x)$. This gives: $y = 8^x - 3$

Part (b): Shifting $f(x)$ 9 units to the left

To shift the function left, we replace $x$ with $x + 9$. This gives: $y = 8^{x + 9}$

Part (c): Reflecting $f(x)$ about the x-axis

To reflect the function across the x-axis, we multiply $f(x)$ by -1. This gives: $y = -8^x$

Summary of Answers:

• (a) $y = 8^x - 3$
• (b) $y = 8^{x + 9}$
• (c) $y = -8^x$

Would you like further details on these transformations or have additional questions?

Here are some related questions that expand on these transformations:

1. How do vertical shifts affect the graph of an exponential function?
2. What happens to the graph of $f(x) = 8^x$ if it's shifted to the right by 5 units?
3. How does a reflection across the y-axis differ from a reflection across the x-axis for exponential functions?
4. What would the equation be if the function $f(x) = 8^x$ is shifted 4 units upward?
5. How does the base of an exponential function affect the steepness of its graph?

Tip: Remember, horizontal transformations affect the $x$-value inside the function, while vertical transformations affect the output (or $y$-value) directly.