Let's break down the transformations one by one based on the given point $(8, -4)$ on the graph of $y = f(x)$.

Part (a): $y = f(x+2) - 1$

• Horizontal shift: The function is shifted to the left by 2 units. This means that the new $x$-value is $8 - 2 = 6$.
• Vertical shift: The graph is shifted downward by 1 unit. This means the new $y$-value is $-4 - 1 = -5$.

Thus, the new point is $(6, -5)$.

Part (b): $y = -2f(x-7) + 9$

• Horizontal shift: The function is shifted to the right by 7 units. This means that the new $x$-value is $8 + 7 = 15$.
• Vertical scaling and shift: First, the function is vertically scaled by a factor of $-2$, so the new $y$-value becomes $-2(-4) = 8$. Then, it is shifted upward by 9 units, so the final $y$-value is $8 + 9 = 17$.

Thus, the new point is $(15, 17)$.

Part (c): $y = f(2x+2)$

• Horizontal scaling and shift: The argument of $f$ is $2x+2$, which can be factored as $2(x+1)$. This represents a horizontal compression by a factor of $\frac{1}{2}$ and a shift to the left by 1 unit. To find the new $x$-value, set $2x + 2 = 8$ (the original $x$-value) and solve for $x$:

$2x + 2 = 8 \quad \Rightarrow \quad 2x = 6 \quad \Rightarrow \quad x = 3$

Thus, the new point is $(3, -4)$, since the $y$-value doesn't change in this transformation.

Summary:

• (a) The new point is $(6, -5)$.
• (b) The new point is $(15, 17)$.
• (c) The new point is $(3, -4)$.

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

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Here are five related questions:

1. How do horizontal shifts affect the graph of a function?
2. What is the impact of vertical scaling by a negative number on a graph?
3. How do you interpret $y = f(2x)$ in terms of graph transformations?
4. What happens when a function is shifted both vertically and horizontally?
5. How do transformations involving multiple operations (like scaling and shifting) work?

Tip: When dealing with transformations, always start by analyzing horizontal changes before applying vertical ones!