Let's address each transformation for the function $f(x) = 9x$:

Part (a): Shift 5 units downward

When a function is shifted downward by $k$ units, we subtract $k$ from $f(x)$. Here, $k = 5$. $y = f(x) - 5 = 9x - 5$

Part (b): Shift 4 units to the right

To shift a function $f(x)$ to the right by $h$ units, we replace $x$ with $x - h$. Here, $h = 4$. $y = f(x - 4) = 9(x - 4) = 9x - 36$

Part (c): Reflect about the y-axis

Reflecting a function $f(x)$ about the y-axis means replacing $x$ with $-x$. $y = f(-x) = 9(-x) = -9x$

Summary

• (a) $y = 9x - 5$
• (b) $y = 9x - 36$
• (c) $y = -9x$

Would you like additional details on any part of these transformations?

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Here are some related questions to explore:

1. How would the equation change if we shifted $f(x) = 9x$ 3 units upward?
2. What is the resulting equation if $f(x) = 9x$ is shifted 2 units to the left?
3. How does reflecting $f(x) = 9x$ about the x-axis affect the equation?
4. What happens to the graph if we stretch $f(x) = 9x$ vertically by a factor of 2?
5. How would you describe the transformations in general terms for any linear function?

Tip: For any function $f(x)$, a shift $h$ units horizontally changes $x$ to $x - h$, and a shift $k$ units vertically adds $k$ to $f(x)$.