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.