Starting with the graph of f(x)=4^x, write the equation of the graph that results from

(a) shifting f(x)f(x) 9 units downward. y=y=

(b) shifting f(x)f(x) 7 units to the right. y=y=

(c) reflecting f(x)f(x) about the y-axis. y=y=

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.

Starting with the graph of f(x)=4^x, write the equation of the graph that results from (a) shifting f(x) 9 units downward. y=y= (b) shifting f(x) 7 units to the right. y=y= (c) reflecting f(x) about the y-axis. y=y=

Explore how to transform the exponential function f(x) = 4^x by shifting it 9 units downward, 7 units to the right, and reflecting it about the y-axis. Learn key concepts in exponential function transformations, including vertical shifts, horizontal shifts, and reflections, suitable for Grades 9-11.

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