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To find the equation of the parabola in the form \( y = ax^2 + bx + c \) that passes through the given points \((-4, -98)\), \((3, -91)\), and \((-1, 1)\), we need to solve for the constants \(a\), \(b\), and \(c\).

### Step 1: Set up a system of equations
Substitute each point into the general form of the quadratic equation.

1. For \((-4, -98)\):  
   \[
   -98 = a(-4)^2 + b(-4) + c \implies -98 = 16a - 4b + c
   \]

2. For \((3, -91)\):  
   \[
   -91 = a(3)^2 + b(3) + c \implies -91 = 9a + 3b + c
   \]

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

### Step 2: Solve the system of equations
We now have the following system:
\[
\begin{aligned}
    16a - 4b + c &= -98 \quad \text{(Equation 1)} \\
    9a + 3b + c &= -91 \quad \text{(Equation 2)} \\
    a - b + c &= 1 \quad \text{(Equation 3)}
\end{aligned}
\]
I'll solve this system to find \(a\), \(b\), and \(c\). Let's proceed with the calculation.The solution to the system of equations gives:

\[
a = -8, \quad b = -7, \quad c = 2
\]

Thus, the equation of the parabola passing through the points \((-4, -98)\), \((3, -91)\), and \((-1, 1)\) is:

\[
y = -8x^2 - 7x + 2
\]

Would you like more details about the solution, or do you have any questions?

Here are five related questions to explore:

1. How can you verify that the points lie on the parabola?
2. What is the vertex of the parabola?
3. Can you derive the axis of symmetry for this quadratic function?
4. How does the discriminant help in understanding the shape of this parabola?
5. How can you express this parabola in vertex form?

**Tip**: To check whether a quadratic equation is correct, substitute the given points back into the equation and ensure that both sides match.

Find an equation in the form y = ax^2 + bx + c for the parabola passing through the points (-4, -98), (3, -91), (-1, 1).

Learn how to find the equation of a parabola in the form y = ax^2 + bx + c that passes through the points (-4, -98), (3, -91), and (-1, 1). This problem involves setting up and solving a system of equations to determine the coefficients a, b, and c. Suitable for students in Grades 9-12.

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