https://file.aiquickdraw.com/mathbot/file/1729912500373wyqlneif.jpg

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?

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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.

Starting with the graph of f(x) = 8^x, write the equation of the graph that results when:
(a) f(x) is shifted 3 units downward.
(b) f(x) is shifted 9 units to the left.
(c) f(x) is reflected about the x-axis.

Learn how to apply graph transformations to the exponential function f(x) = 8^x, including vertical shifts, horizontal shifts, and reflections across the x-axis. This guide explains each transformation with equations and examples, suitable for students in Grades 10-12.

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