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.