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This series of problems will build your fluency with recognizing and finding equivalent forms of quadratic functions. For each question below, match each equivalent form by circling them in the same color. Some sets might have a graph, factored equation, and standard form equation, while other sets might only have 2 elements. Some may have no matches. 

The image provided shows a series of quadratic functions in different forms (factored form, standard form, and graphs). The task requires matching equivalent forms of each quadratic function, which may include matching a factored equation, a standard form equation, and/or a graph.

Let me go through the image step by step:

### Equations and Graphs:
1. **A. \( f(x) = (x + 1)(x - 1) \)** (Factored form)
   - This simplifies to \( f(x) = x^2 - 1 \) (Standard form).
   - The corresponding graph will be a parabola opening upwards, with roots at \( x = -1 \) and \( x = 1 \) and a vertex at \( (0, -1) \).
   - Matching graph: **H.** (This is a graph of \( f(x) = x^2 - 1 \)).

2. **E. \( f(x) = (x)(x + 3) \)** (Factored form)
   - This simplifies to \( f(x) = x^2 + 3x \) (Standard form).
   - The corresponding graph will be a parabola opening upwards, with a root at \( x = 0 \) and \( x = -3 \), and the vertex between these points.
   - Matching graph: **B.** (This graph shows a parabola with roots at \( x = 0 \) and \( x = -3 \)).

3. **I. \( f(x) = x^2 - 9 \)** (Standard form)
   - This factors as \( f(x) = (x - 3)(x + 3) \) (Factored form).
   - The corresponding graph will be a parabola opening upwards, with roots at \( x = -3 \) and \( x = 3 \), and the vertex at \( (0, -9) \).
   - Matching graph: **F.** (This graph has roots at \( x = -3 \) and \( x = 3 \), with a vertex at \( (0, -9) \)).

4. **C. \( f(x) = x^2 - 6x + 9 \)** (Standard form)
   - This factors as \( f(x) = (x - 3)^2 \) (Factored form).
   - The graph will be a parabola with a double root at \( x = 3 \), and the vertex at \( (3, 0) \).
   - Matching graph: **U.** (This graph shows a parabola with a vertex at \( (3, 0) \) and no crossing, indicating a double root).

5. **G. \( f(x) = x^2 - 1 \)** (Standard form)
   - This factors as \( f(x) = (x - 1)(x + 1) \) (Factored form, same as Equation A).
   - Matching graph: **H.** (This is the same graph as matched for A).

6. **K. \( f(x) = (x - 1)^2 \)** (Factored form)
   - This expands to \( f(x) = x^2 - 2x + 1 \) (Standard form).
   - The graph will be a parabola with a vertex at \( (1, 0) \), indicating a double root at \( x = 1 \).
   - Matching graph: **I.** (This shows a parabola with a vertex at \( (1, 0) \)).

### Summary of Matches:
- **A** and **G** match with **H**.
- **E** matches with **B**.
- **I** matches with **F**.
- **C** matches with **U**.
- **K** matches with **I**.

Would you like a detailed explanation of any of these matches? Let me know if you have any questions!

Here are 5 questions to think about:
1. What are the key differences between factored and standard form for quadratic functions?
2. How do the roots of a quadratic relate to its graph?
3. How does the vertex form of a quadratic help in understanding the graph’s transformations?
4. How can you recognize a double root from both an equation and its graph?
5. Why are factored forms helpful for identifying roots easily?

**Tip:** Always check the signs in factored form equations to correctly identify the location of the roots in the graph.

This series of problems will build your fluency with recognizing and finding equivalent forms of quadratic functions. For each question below, match each equivalent form by circling them in the same color. Some sets might have a graph, factored equation, and standard form equation, while other sets might only have 2 elements. Some may have no matches.

This problem set involves matching quadratic functions in different forms: standard form, factored form, and graphical representation. Students will identify and circle equivalent forms of these functions, applying key algebraic concepts and graphing techniques. Suitable for Grades 9-10.

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