Find the equation of the parabola passing through (1,0),(2,12) and (-2,-12)

To find the equation of a parabola passing through three points, we can use the general form of the equation of a parabola:

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

Given points: \((1, 0)\), \((2, 12)\), and \((-2, -12)\).

We'll substitute each point into the equation to form a system of linear equations.

### Step 1: Substitute the points into the parabola equation

1. For \((1, 0)\):
   \[
   0 = a(1)^2 + b(1) + c \implies a + b + c = 0
   \]

2. For \((2, 12)\):
   \[
   12 = a(2)^2 + b(2) + c \implies 4a + 2b + c = 12
   \]

3. For \((-2, -12)\):
   \[
   -12 = a(-2)^2 + b(-2) + c \implies 4a - 2b + c = -12
   \]

### Step 2: Solve the system of equations

We have the following system of equations:
1. \( a + b + c = 0 \)
2. \( 4a + 2b + c = 12 \)
3. \( 4a - 2b + c = -12 \)

Let's subtract equation 1 from equations 2 and 3 to eliminate \( c \):

- Subtract equation 1 from equation 2:
  \[
  (4a + 2b + c) - (a + b + c) = 12 - 0
  \]
  \[
  3a + b = 12 \quad \text{(Equation 4)}
  \]

- Subtract equation 1 from equation 3:
  \[
  (4a - 2b + c) - (a + b + c) = -12 - 0
  \]
  \[
  3a - 3b = -12 \quad \text{(Equation 5)}
  \]

### Step 3: Solve for \( a \) and \( b \)

From Equation 5:
\[
a - b = -4 \quad \text{(Divide by 3)}
\]
\[
b = a + 4 \quad \text{(Equation 6)}
\]

Substitute Equation 6 into Equation 4:
\[
3a + (a + 4) = 12
\]
\[
4a + 4 = 12
\]
\[
4a = 8
\]
\[
a = 2
\]

Substitute \( a = 2 \) into Equation 6:
\[
b = 2 + 4 = 6
\]

### Step 4: Solve for \( c \)

Using \( a = 2 \) and \( b = 6 \) in the first original equation:
\[
2 + 6 + c = 0
\]
\[
c = -8
\]

### Final Equation of the Parabola
The equation of the parabola is:

\[
\boxed{y = 2x^2 + 6x - 8}
\]

Would you like more details on this process or have any questions?

### 5 Related Questions
1. How can you verify that the parabola passes through the given points?
2. What are the key features (vertex, axis of symmetry, etc.) of the parabola \( y = 2x^2 + 6x - 8 \)?
3. How do you find the focus and directrix of a parabola given its equation?
4. What is the significance of the discriminant in determining the nature of the roots of a quadratic equation?
5. How can you derive the vertex form of a parabola from the standard form equation?

### Tip
Always double-check your algebraic manipulations when solving systems of equations to avoid errors in the final result.

Learn how to find the equation of a parabola passing through specific points (1, 0), (2, 12), and (-2, -12) with a step-by-step solution. This problem involves using the general form of a parabola and solving a system of linear equations. Suitable for high school students studying quadratic equations.

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