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The given function is \( f(x) = 2^x \), and we are asked to find the equations of the transformed graphs under specific transformations. Let’s go through each part:

### (a) \( f(x) \) is shifted 5 units downward.

When a function is shifted downward, we subtract the shift value from the entire function. Therefore, the equation becomes:
\[
y = 2^x - 5
\]

### (b) \( f(x) \) is shifted 6 units to the right.

When a function is shifted to the right, we replace \( x \) with \( x - h \), where \( h \) is the number of units shifted. Here, \( h = 6 \), so the equation becomes:
\[
y = 2^{x - 6}
\]

### (c) \( f(x) \) is reflected about the y-axis.

To reflect a function about the y-axis, we replace \( x \) with \( -x \). Therefore, the equation becomes:
\[
y = 2^{-x}
\]

### Summary of Equations:

- (a) \( y = 2^x - 5 \)
- (b) \( y = 2^{x - 6} \)
- (c) \( y = 2^{-x} \)

Would you like more details or further explanations on any part?

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Here are 5 related questions to explore:
1. How does shifting a graph leftward differ from shifting it rightward?
2. How would the graph change if \( f(x) = 2^x \) were shifted upward by 7 units?
3. What happens if the graph is reflected about the x-axis instead of the y-axis?
4. How would the equation change if the graph were shifted 3 units to the left and 4 units upward?
5. Can the same transformation rules be applied to logarithmic functions like \( \log(x) \)?

**Tip:** When reflecting across the y-axis, pay attention to the sign of the exponent in exponential functions. This reverses the direction of the graph.

Starting with the graph of f(x) = 2^x, write the equation of the graph that results when: (a) f(x) is shifted 5 units downward, (b) f(x) is shifted 6 units to the right, (c) f(x) is reflected about the y-axis.

Explore how to transform the exponential function f(x) = 2^x through vertical shifts, horizontal shifts, and reflections. This detailed breakdown covers shifts of 5 units downward, 6 units to the right, and reflection about the y-axis. Suitable for students in Grades 9-11.

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