Let's analyze the given transformations step by step. The original function is:

$f(x) = 3x$

(a) Shift 5 units upward:

To shift a function upward by $c$ units, you add $c$ to the function. In this case, the shift is 5 units upward, so the new function is:

$y = 3x + 5$

(b) Shift 8 units to the left:

To shift a function to the left by $c$ units, you replace $x$ with $x + c$ in the function. Since the shift is 8 units to the left, you substitute $x$ with $x + 8$, so the new function is:

$y = 3(x + 8)$

Expanding this:

$y = 3x + 24$

(c) Reflection about the x-axis and y-axis:

• Reflection about the x-axis: Multiply the function by $-1$.
• Reflection about the y-axis: Replace $x$ with $-x$.

Applying both transformations to the original function $f(x) = 3x$:

1. Reflecting about the x-axis gives $-3x$.
2. Reflecting about the y-axis gives $-3(-x) = 3x$.

Combining both reflections, we get:

$y = -3(-x) = 3x$

However, reflecting about both axes will result in the original function since it gets reversed twice.

Therefore, after reflecting about both axes, the equation remains:

$y = -3x$

Summary of results:

• (a) Shift 5 units upward: $y = 3x + 5$
• (b) Shift 8 units to the left: $y = 3x + 24$
• (c) Reflected about the x-axis and y-axis: $y = -3x$

Would you like more details or clarification?

Here are 5 related questions to explore further:

1. How would the equation change if the function were shifted 5 units downward instead of upward?
2. What would happen if the function were shifted 8 units to the right instead of to the left?
3. How do you reflect a function about the y-axis alone?
4. How does the slope change under different transformations?
5. What is the effect of combining shifts and reflections?

Tip: When applying transformations to a graph, always think of shifts as changes to the inputs (horizontal) or outputs (vertical) and reflections as flipping the graph across an axis.