Let's solve each part based on the original function $f(x) = 4^x$:

(a) Shift $f(x)$ 6 units upwards

Shifting a function $f(x)$ vertically by adding a constant $k$ translates the graph $k$ units up. Therefore, shifting $f(x) = 4^x$ 6 units upwards gives:

$y = 4^x + 6$

(b) Shift $f(x)$ 8 units to the left

Shifting a function horizontally by $h$ units to the left involves replacing $x$ with $x + h$. Shifting $f(x) = 4^x$ 8 units to the left results in:

$y = 4^{x + 8}$

(c) Reflect $f(x)$ about the $x$-axis

Reflecting a function about the $x$-axis involves multiplying the function by $-1$. Therefore, reflecting $f(x) = 4^x$ about the $x$-axis gives:

$y = -4^x$

Would you like further details on any of these transformations, or have additional questions?

Here are five related questions to expand on this topic:

1. What happens to the graph of $f(x) = 4^x$ when shifted down by 6 units?
2. How does shifting a function horizontally affect its domain?
3. What is the effect of reflecting a function about the $y$-axis instead of the $x$-axis?
4. How does the equation change if $f(x) = 4^x$ is stretched vertically by a factor of 3?
5. What is the new equation if $f(x) = 4^x$ is shifted 5 units to the right?

Tip: When transforming functions, remember that vertical changes affect the output directly, while horizontal changes alter the input variable.