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!