Question

If the points in the table lie on a​ parabola, write the equation whose graph is the parabola.
x
negative 1
1
3
5
y
negative 7
13
negative 7
negative 67
Question content area bottom
Part 1
yequals
  
enter your response here

Answer 1

To find the equation of a parabola, we can generally express it in the form:

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

Given the points:
- \((-1, -7)\)
- \((1, 13)\)
- \((3, -7)\)
- \((5, -67)\)

We will use three of these points to create a system of equations.

### Step 1: Set up equations using three points

1. **For the point \((-1, -7)\)**:
\[
-7 = a(-1)^2 + b(-1) + c \implies -7 = a - b + c \quad \text{(Equation 1)}
\]

2. **For the point \((1, 13)\)**:
\[
13 = a(1)^2 + b(1) + c \implies 13 = a + b + c \quad \text{(Equation 2)}
\]

3. **For the point \((3, -7)\)**:
\[
-7 = a(3)^2 + b(3) + c \implies -7 = 9a + 3b + c \quad \text{(Equation 3)}
\]

### Step 2: Solve the system of equations

Now we have the following system:

1. \( a - b + c = -7 \)
2. \( a + b + c = 13 \)
3. \( 9a + 3b + c = -7 \)

We can solve this system step by step.

### Step 3: Eliminate \(c\)

Subtract Equation 1 from Equation 2:

\[
(a + b + c) - (a - b + c) = 13 - (-7)
\]
\[
2b = 20 \implies b = 10
\]

Now substitute \(b = 10\) into Equations 1 and 2 to find \(a\) and \(c\).

Using Equation 1:

\[
a - 10 + c = -7 \implies a + c = 3 \quad \text{(Equation 4)}
\]

Using Equation 2:

\[
a + 10 + c = 13 \implies a + c = 3 \quad \text{(Confirmed)}
\]

Now substitute \(b = 10\) into Equation 3:

\[
9a + 3(10) + c = -7 \implies 9a + 30 + c = -7
\]
\[
9a + c = -37 \quad \text{(Equation 5)}
\]

### Step 4: Solve Equations 4 and 5

Now we have:

1. \( a + c = 3 \) (Equation 4)
2. \( 9a + c = -37 \) (Equation 5)

Subtract Equation 4 from Equation 5:

\[
(9a + c) - (a + c) = -37 - 3
\]
\[
8a = -40 \implies a = -5
\]

Now substitute \(a = -5\) back into Equation 4 to find \(c\):

\[
-5 + c = 3 \implies c = 8
\]

### Summary of coefficients

Now we have:

- \(a = -5\)
- \(b = 10\)
- \(c = 8\)

So, the equation of the parabola is:

\[
y = -5x^2 + 10x + 8
\]

### Final Answer

The equation of the parabola is:

\[
y = -5x^2 + 10x + 8
\]