Question

If the points in the table lie on a​ parabola, write the equation whose graph is the parabola.
x
negative 1−1
11
33
55
y
negative 6−6
1414
negative 6−6
negative 66−66
Question content area bottom
Part 1
yequals= enter your response here
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)

Answer 1

To find the equation of the parabola that fits the points given in the table, we can use the standard form of a quadratic equation:

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

The points provided are:

1. (-1, -6)
2. (1, 14)
3. (3, -6)
4. (5, -6)

Let's assign the coordinates for these points:

1. \((-1, -6) \rightarrow (-1, -6)\)
2. \((1, 14) \rightarrow (1, 14)\)
3. \((3, -6) \rightarrow (3, -6)\)
4. \((5, -6) \rightarrow (5, -6)\)

Since we have four points, we can use three of them to set up a system of equations.

Using points 1, 2, and 3:

1. For \((-1, -6)\):
\[
-6 = a(-1)^2 + b(-1) + c \implies -6 = a - b + c \quad \text{(Equation 1)}
\]

2. For \((1, 14)\):
\[
14 = a(1)^2 + b(1) + c \implies 14 = a + b + c \quad \text{(Equation 2)}
\]

3. For \((3, -6)\):
\[
-6 = a(3)^2 + b(3) + c \implies -6 = 9a + 3b + c \quad \text{(Equation 3)}
\]

Now we'll write these equations in matrix form and solve for \(a\), \(b\), and \(c\).

### Step 1: Set Up the System

From our equations:
- Equation 1: \(a - b + c = -6\)
- Equation 2: \(a + b + c = 14\)
- Equation 3: \(9a + 3b + c = -6\)

### Step 2: Subtract Equations

Let's eliminate \(c\) by subtracting Equation 1 from Equation 2 and then Equation 2 from Equation 3:

From Equation 1 and Equation 2:
\[
(a + b + c) - (a - b + c) = 14 - (-6) \\
2b = 20 \implies b = 10
\]

From Equation 2 and Equation 3:
\[
(9a + 3b + c) - (a + b + c) = -6 - 14 \\
8a + 2b = -20 \implies 8a + 2(10) = -20 \\
8a + 20 = -20 \\
8a = -40 \implies a = -5
\]

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

Now substitute \(a\) and \(b\) back into Equation 1 to find \(c\):
\[
-5 - 10 + c = -6 \\
c = -6 + 15 = 9
\]

### Conclusion

Now we have the values:
- \(a = -5\)
- \(b = 10\)
- \(c = 9\)

Thus, the equation of the parabola is:
\[
y = -5x^2 + 10x + 9
\]

This is your final answer. You can enter it as:

\[ y = -5x^2 + 10x + 9 \]