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?

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.