https://file.aiquickdraw.com/mathbot/file/1726014727755wrkrql81.jpg

To match the equations with the graphs, let's analyze both the graphs and the equations:

### Given Equations:
- **A. \( f(x) = (x + 4)^2 - 1 \)**: This is a parabola with its vertex at \( (-4, -1) \). Since \( (x + 4) \) shifts the graph left by 4 units and \( -1 \) shifts it down by 1 unit.
- **C. \( f(x) = (x - 4)^2 - 1 \)**: This parabola has its vertex at \( (4, -1) \), shifted right by 4 units and down by 1 unit.
- **E. \( f(x) = 2(x - 4)^2 - 1 \)**: This parabola also has its vertex at \( (4, -1) \), but is vertically stretched because of the coefficient 2, making it narrower compared to the others.

### Graphs:
- **Graph B**: This graph has its vertex at \( (-4, -1) \) and opens upwards. This corresponds to Equation **A** \( f(x) = (x + 4)^2 - 1 \).
- **Graph D**: This graph has its vertex at \( (4, -1) \) and opens upwards without any stretch. This corresponds to Equation **C** \( f(x) = (x - 4)^2 - 1 \).
- **Graph F**: This graph has its vertex at \( (4, -1) \) but is narrower, indicating a vertical stretch. This corresponds to Equation **E** \( f(x) = 2(x - 4)^2 - 1 \).

### Final Matches:
- **A. \( f(x) = (x + 4)^2 - 1 \)** matches with **Graph B**.
- **C. \( f(x) = (x - 4)^2 - 1 \)** matches with **Graph D**.
- **E. \( f(x) = 2(x - 4)^2 - 1 \)** matches with **Graph F**.

Would you like more details on how transformations of parabolas work, or do you have any further questions?

Here are five related questions for further practice:
1. How does changing the sign of the coefficient of \( x^2 \) affect the parabola?
2. What happens to the parabola when the constant term is increased or decreased?
3. How does the value of the coefficient in front of \( (x - h)^2 \) affect the width of the parabola?
4. How would the graph change if the equation was \( f(x) = (x - 4)^2 + 3 \)?
5. Can you determine the axis of symmetry for each of these equations?

**Tip**: The vertex form of a quadratic equation, \( f(x) = a(x - h)^2 + k \), makes it easy to identify the vertex as \( (h, k) \) and the factor \( a \) controls the width and direction of the parabola.

Match the graphs and equations provided in the image.

This exercise helps students in Grades 9-10 match quadratic function equations to their corresponding graphs. It covers key algebraic concepts such as the vertex form of a quadratic function and graph transformations.

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