Write an equation of the parabola that passes through the points (1,–
8), (–
1,10), and (–
2,13).

To find the equation of the parabola passing through the points \((1, -8)\), \((-1, 10)\), and \((-2, 13)\), we start by assuming the general equation of a parabola in standard form:

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

We will use the coordinates of the given points to create a system of equations. Each point \((x, y)\) satisfies the parabola equation, so we substitute the values of \(x\) and \(y\) for each point into the equation.

1. For \((1, -8)\):
\[
-8 = a(1)^2 + b(1) + c \quad \Rightarrow \quad a + b + c = -8
\]

2. For \((-1, 10)\):
\[
10 = a(-1)^2 + b(-1) + c \quad \Rightarrow \quad a - b + c = 10
\]

3. For \((-2, 13)\):
\[
13 = a(-2)^2 + b(-2) + c \quad \Rightarrow \quad 4a - 2b + c = 13
\]

Now we have the following system of equations:

\[
\begin{aligned}
a + b + c &= -8 \\
a - b + c &= 10 \\
4a - 2b + c &= 13
\end{aligned}
\]

We will solve this system step by step to find \(a\), \(b\), and \(c\).

### Step 1: Eliminate one variable
We can subtract the second equation from the first to eliminate \(b\):

\[
(a + b + c) - (a - b + c) = -8 - 10 \quad \Rightarrow \quad 2b = -18 \quad \Rightarrow \quad b = -9
\]

### Step 2: Substitute \(b = -9\) into the first and second equations
Substitute \(b = -9\) into the first equation:
\[
a - 9 + c = -8 \quad \Rightarrow \quad a + c = 1 \quad \text{(Equation 4)}
\]

Substitute \(b = -9\) into the second equation:
\[
a + 9 + c = 10 \quad \Rightarrow \quad a + c = 1 \quad \text{(same as Equation 4)}
\]

### Step 3: Substitute \(b = -9\) into the third equation
Substitute \(b = -9\) into the third equation:
\[
4a - 2(-9) + c = 13 \quad \Rightarrow \quad 4a + 18 + c = 13 \quad \Rightarrow \quad 4a + c = -5 \quad \text{(Equation 5)}
\]

Now we have two equations:
\[
a + c = 1 \quad \text{(Equation 4)}
\]
\[
4a + c = -5 \quad \text{(Equation 5)}
\]

### Step 4: Solve for \(a\) and \(c\)
Subtract Equation 4 from Equation 5:
\[
(4a + c) - (a + c) = -5 - 1 \quad \Rightarrow \quad 3a = -6 \quad \Rightarrow \quad a = -2
\]

Now substitute \(a = -2\) into Equation 4:
\[
-2 + c = 1 \quad \Rightarrow \quad c = 3
\]

### Final Equation
We have \(a = -2\), \(b = -9\), and \(c = 3\). Therefore, the equation of the parabola is:

\[
y = -2x^2 - 9x + 3
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How can we derive the equation of a parabola from its vertex form?
2. What role does each coefficient in the quadratic equation play in the graph's shape?
3. How can we use the discriminant to determine the number of roots of a parabola?
4. How would you find the axis of symmetry for this parabola?
5. What is the general method to find the vertex of a parabola given its equation?

**Tip:** When solving systems of equations, eliminating one variable at a time simplifies the process greatly!

Write an equation of the parabola that passes through the points (1, -8), (-1, 10), and (-2, 13).

Solve for the equation of a parabola passing through the points (1, -8), (-1, 10), and (-2, 13). This detailed step-by-step solution covers concepts in quadratic equations and solving systems of equations using substitution and elimination methods. Ideal for students in Grades 9-11.

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