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The graph represents a quadratic curve, and you are asked to find the equation of the curve in the form:

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

We are given the following key points from the graph:
1. The x-intercepts are \( x = -1 \) and \( x = 3 \), which gives the roots of the quadratic equation.
2. The vertex is at \( (1, -9) \).

### Step 1: Form the general equation
Since the quadratic equation has roots at \( x = -1 \) and \( x = 3 \), we can write it in its factored form as:

\[
y = a(x + 1)(x - 3)
\]

Here, \( a \) is a constant that needs to be determined. 

### Step 2: Use the vertex to find \( a \)
The vertex of the parabola is given as \( (1, -9) \), meaning when \( x = 1 \), \( y = -9 \). We can substitute \( x = 1 \) and \( y = -9 \) into the equation to find the value of \( a \).

\[
-9 = a(1 + 1)(1 - 3)
\]
\[
-9 = a(2)(-2)
\]
\[
-9 = -4a
\]
\[
a = \frac{-9}{-4} = \frac{9}{4}
\]

Thus, the equation of the quadratic becomes:

\[
y = \frac{9}{4}(x + 1)(x - 3)
\]

### Step 3: Expand the equation
Now, expand the equation to get it in the form \( y = ax^2 + bx + c \).

\[
y = \frac{9}{4}(x^2 - 3x + x - 3)
\]
\[
y = \frac{9}{4}(x^2 - 2x - 3)
\]
\[
y = \frac{9}{4}x^2 - \frac{18}{4}x - \frac{27}{4}
\]

Finally, we multiply through by 4 to clear the fractions:

\[
4y = 9x^2 - 18x - 27
\]

So, the equation of the curve is:

\[
y = \frac{9}{4}x^2 - \frac{18}{4}x - \frac{27}{4}
\]




Find the equation of the quadratic curve given its graph with x-intercepts at x = -1 and x = 3, and a vertex at (1, -9).

Learn how to determine the equation of a quadratic curve given its x-intercepts at -1 and 3, and a vertex at (1, -9). This solution explains the step-by-step process of forming the equation in factored form and using the vertex to solve for the constant.

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