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Find a quadratic function to model the values in the table.

To find a quadratic function that models the values in the table, we will assume the general form of a quadratic function:

\[
y = ax^2 + bx + c
\]

We will use the points provided in the table to create a system of equations and solve for \(a\), \(b\), and \(c\).

### Points from the table:
1. \((-1, -3)\)
2. \((0, -3)\)
3. \((3, 33)\)

We can substitute these points into the quadratic function:

1. For \((-1, -3)\):
\[
-3 = a(-1)^2 + b(-1) + c \Rightarrow -3 = a - b + c
\]

2. For \((0, -3)\):
\[
-3 = a(0)^2 + b(0) + c \Rightarrow -3 = c
\]

3. For \((3, 33)\):
\[
33 = a(3)^2 + b(3) + c \Rightarrow 33 = 9a + 3b + c
\]

From equation 2, we know that \(c = -3\). Now, we can substitute \(c = -3\) into the other two equations:

1. \( -3 = a - b - 3 \Rightarrow a - b = 0 \Rightarrow a = b \)
2. \( 33 = 9a + 3b - 3 \Rightarrow 36 = 9a + 3b \Rightarrow 36 = 12a \Rightarrow a = 3 \)

Since \(a = b\), we also have \(b = 3\).

Thus, the quadratic function is:

\[
y = 3x^2 + 3x - 3
\]

This is the quadratic function that models the values in the table.

Would you like more details on solving this system, or do you have any other questions? 

Here are 5 related questions you might find helpful:

1. How do you solve a system of equations using substitution or elimination?
2. What is the significance of each term in a quadratic function \(y = ax^2 + bx + c\)?
3. How can you determine the vertex of a quadratic function?
4. What is the role of the discriminant in a quadratic equation?
5. How can quadratic functions model real-world scenarios?

**Tip:** When modeling with quadratics, always try to use points that cover a range of values for \(x\) to ensure an accurate fit.

Learn how to find a quadratic function to model the values in a data table. This problem covers algebraic concepts, including quadratic functions and solving systems of equations using substitution. Suitable for students in Grades 9-11.

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