https://file.aiquickdraw.com/mathbot/file/1728541129194dj0xb54a.jpg

We are given the graph of a quadratic function, and we need to find its equation in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are integers.

From the graph, we can observe the following important points:
- The parabola crosses the x-axis at \( x = -1 \) and \( x = 3 \), meaning these are the roots of the equation.
- The vertex of the parabola occurs at \( (1, -9) \), as this is the lowest point of the curve.

### Step 1: General form of the quadratic equation
The fact that the roots of the quadratic are \( x = -1 \) and \( x = 3 \) allows us to write the quadratic in its factored form:
\[
y = a(x + 1)(x - 3)
\]
where \( a \) is a constant that controls the vertical stretch of the parabola.

### Step 2: Use the vertex to find \( a \)
We know the vertex is \( (1, -9) \), so we can substitute \( x = 1 \) and \( y = -9 \) into the equation to find \( a \).

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

### Step 3: Rewrite the equation
Thus, the equation of the quadratic is:
\[
y = \frac{9}{4}(x + 1)(x - 3)
\]

But we need to express the equation in standard form, \( y = ax^2 + bx + c \), and with integer coefficients. So, let's expand and simplify.

### Step 4: Expand and simplify
First, expand \( (x + 1)(x - 3) \):
\[
(x + 1)(x - 3) = x^2 - 3x + x - 3 = x^2 - 2x - 3
\]

Now multiply by \( \frac{9}{4} \):
\[
y = \frac{9}{4}(x^2 - 2x - 3)
\]
\[
y = \frac{9}{4}x^2 - \frac{18}{4}x - \frac{27}{4}
\]

Finally, to get rid of the fractions, multiply through by 4:
\[
4y = 9x^2 - 18x - 27
\]
\[
y = \frac{9x^2 - 18x - 27}{4}
\]

Thus the equation is \( y = 

Find the equation of the curve, giving your answer in the form y = ax^2 + bx + c where a, b, and c are integers.

Learn how to find the equation of a quadratic curve using its x-intercepts and vertex. This problem demonstrates key concepts of algebra, including the factored form and the vertex of a parabola, and is suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation