Let's analyze each part of the question regarding the transformations on the function $f(x) = 5^x$.

(a) Shifting $f(x)$ 1 unit upward

To shift a function upward by $k$ units, we add $k$ to the function. In this case, $k = 1$.

• New function: $y = f(x) + 1 = 5^x + 1$.

(b) Shifting $f(x)$ 8 units to the right

To shift a function to the right by $h$ units, we replace $x$ with $x - h$ in the function. Here, $h = 8$.

• New function: $y = f(x - 8) = 5^{x - 8}$.

(c) Reflecting $f(x)$ about the x-axis and the y-axis

1. Reflection about the x-axis: This changes $f(x)$ to $-f(x)$.
2. Reflection about the y-axis: This changes $f(x)$ to $f(-x)$.

When reflecting about both axes, we apply both transformations:

• New function: $y = -f(-x) = -5^{-x}$.

Summary of Results

• (a) $y = 5^x + 1$
• (b) $y = 5^{x - 8}$
• (c) $y = -5^{-x}$

Would you like further explanation on any part of these transformations?

Here are five related questions to deepen your understanding:

1. How would you shift $f(x) = 5^x$ down by 3 units?
2. What would be the result of shifting $f(x) = 5^x$ 5 units to the left?
3. How does reflecting $f(x) = 5^x$ only about the y-axis change its equation?
4. What does a vertical stretch look like for $f(x) = 5^x$?
5. How would the graph change if we applied both upward and downward shifts successively?

Tip: Transformations applied to functions often follow similar rules, regardless of the type of function (e.g., linear, exponential). Remembering the basics of shifting and reflecting can help you handle any function transformation.