Find an equation in the form y equals ax squared plus bx plus c for the parabola passing through the points. ​(1​,12​), ​(negative 3​,64​), ​(negative 2​,24​)

To find the equation of the parabola in the form \( y = ax^2 + bx + c \) that passes through the given points \((1, 12)\), \((-3, 64)\), and \((-2, 24)\), we can set up a system of equations by substituting each point into the equation.

### Step 1: Substitute each point into the equation \( y = ax^2 + bx + c \).

For \((1, 12)\):
\[
12 = a(1)^2 + b(1) + c \implies a + b + c = 12
\]

For \((-3, 64)\):
\[
64 = a(-3)^2 + b(-3) + c \implies 9a - 3b + c = 64
\]

For \((-2, 24)\):
\[
24 = a(-2)^2 + b(-2) + c \implies 4a - 2b + c = 24
\]

### Step 2: Write the system of equations.

\[
\begin{aligned}
1) & \quad a + b + c = 12 \\
2) & \quad 9a - 3b + c = 64 \\
3) & \quad 4a - 2b + c = 24
\end{aligned}
\]

### Step 3: Solve the system of equations.

Subtract the first equation from the second and third equations to eliminate \( c \):

\[
(9a - 3b + c) - (a + b + c) = 64 - 12
\]
\[
8a - 4b = 52 \implies 2a - b = 13 \quad \text{(Equation 4)}
\]

\[
(4a - 2b + c) - (a + b + c) = 24 - 12
\]
\[
3a - 3b = 12 \implies a - b = 4 \quad \text{(Equation 5)}
\]

Now, subtract Equation 5 from Equation 4 to eliminate \( b \):

\[
(2a - b) - (a - b) = 13 - 4
\]
\[
a = 9
\]

Substitute \( a = 9 \) into Equation 5:

\[
9 - b = 4 \implies b = 5
\]

Finally, substitute \( a = 9 \) and \( b = 5 \) into the first equation:

\[
9 + 5 + c = 12 \implies c = -2
\]

### Step 4: Write the final equation.

The equation of the parabola is:
\[
\boxed{y = 9x^2 + 5x - 2}
\]

Would you like a detailed explanation on any step or have any further questions?

### Related Questions:
1. How do you verify if the parabola equation passes through given points?
2. What is the significance of each coefficient \( a \), \( b \), and \( c \) in the equation of a parabola?
3. How can you derive the vertex of the parabola from the equation?
4. How does changing the value of \( a \) affect the shape of the parabola?
5. Can this method be extended to parabolas that open sideways?

### Tip:
When solving systems of equations, it's often helpful to eliminate one variable at a time to simplify the process.

Find an equation in the form y equals ax squared plus bx plus c for the parabola passing through the points (1,12), (negative 3,64), (negative 2,24).

Learn how to find the quadratic equation y = ax^2 + bx + c for a parabola passing through the points (1,12), (-3,64), and (-2,24). This step-by-step guide covers key algebraic concepts like solving systems of equations, suitable for students in Grades 9-11.

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