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Let's solve the transformations one by one based on the function \( f(x) = 5^x \):

### (a) Shifting \( f(x) \) 6 units upward:
- When a function is shifted vertically upward by \( k \) units, the transformation is \( y = f(x) + k \).
- In this case, \( k = 6 \).
- Therefore, the equation becomes:
  \[
  y = 5^x + 6
  \]

### (b) Shifting \( f(x) \) 8 units to the right:
- When a function is shifted horizontally to the right by \( h \) units, the transformation is \( y = f(x - h) \).
- In this case, \( h = 8 \).
- Therefore, the equation becomes:
  \[
  y = 5^{x - 8}
  \]

### (c) Reflecting \( f(x) \) about the x-axis:
- Reflecting a function across the x-axis changes the sign of the function. The transformation is \( y = -f(x) \).
- Therefore, the equation becomes:
  \[
  y = -5^x
  \]

### (d) Reflecting \( f(x) \) about the line \( y = 4 \):
- To reflect a function over a line \( y = c \), we perform the transformation \( y = 2c - f(x) \).
- In this case, \( c = 4 \).
- Therefore, the equation becomes:
  \[
  y = 2(4) - 5^x = 8 - 5^x
  \]

Let me know if you'd like further details or have any questions!

Here are 5 related questions to expand your understanding:
1. How would you shift the graph of \( f(x) = 5^x \) 10 units downward?
2. What happens if you reflect \( f(x) = 5^x \) across the y-axis?
3. How would the graph change if you applied both a horizontal and vertical shift?
4. What would the equation be if you shifted \( f(x) = 5^x \) 4 units to the left?
5. How does the steepness of \( f(x) = 5^x \) compare to other exponential functions like \( f(x) = 2^x \)?

**Tip:** Reflections and shifts can be combined for more complex transformations. Make sure to apply each transformation in the correct order!

Starting with the graph of f(x) = 5^x, write the equation of the graph that results from (a) shifting f(x) 6 units upward, (b) shifting f(x) 8 units to the right, (c) reflecting f(x) about the x-axis, and (d) reflecting f(x) about the line y = 4.

Explore how to shift and reflect the graph of the exponential function f(x) = 5^x. Learn the effects of vertical and horizontal shifts, reflections across the x-axis, and reflections about a line. This problem involves key graph transformation formulas, suitable for high school students.

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