Suppose left parenthesis 8 comma negative 4 right parenthesis(8,−4) is a point on the graph of yequals=​f(x).
​(a) What point is on the graph of y equals f left parenthesis x plus 2 right parenthesis minus 1y=f(x+2)−1​?
​(b) What point is on the graph of y equals negative 2 f left parenthesis x minus 7 right parenthesis plus 9y=−2f(x−7)+9​?
​(c) What point is on the graph of y equals f left parenthesis 2 x plus 2 right parenthesisy=f(2x+2)​?

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!

Suppose (8,−4) is a point on the graph of y=f(x). (a) What point is on the graph of y=f(x+2)−1? (b) What point is on the graph of y=−2f(x−7)+9? (c) What point is on the graph of y=f(2x+2)?

Explore how transformations affect the graph of a function with step-by-step solutions for points on y=f(x+2)−1, y=−2f(x−7)+9, and y=f(2x+2). Learn about horizontal and vertical shifts, scaling, and how to apply them. Suitable for high school students in Grades 9-12.

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