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
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)

Answer 1

To find the equation of the parabola given the points from the table, we can use the general form of a quadratic equation:

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

We have four points to consider:
- (-1, -7)
- (1, 13)
- (3, -7)
- (5, -67)

Since there are three coefficients \(a\), \(b\), and \(c\), we can derive three equations from three of the points, and then we can test for a fourth point to ensure our solution is correct.

Let's use the first three points. We can set up the following equations:

1. From (-1, -7):
\[
-7 = a(-1)^2 + b(-1) + c \implies -7 = a - b + c \quad (1)
\]

2. From (1, 13):
\[
13 = a(1)^2 + b(1) + c \implies 13 = a + b + c \quad (2)
\]

3. From (3, -7):
\[
-7 = a(3)^2 + b(3) + c \implies -7 = 9a + 3b + c \quad (3)
\]

We now have three equations:

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

Now let's solve this system of equations:

**Step 1**: Subtract equations (1) from (2):
\[
(a + b + c) - (a - b + c) = 13 + 7 \implies 2b = 20 \implies b = 10
\]

**Step 2**: Substitute \(b = 10\) into equations (1) and (2):

Substituting into (1):
\[
a - 10 + c = -7 \implies a + c = 3 \quad (4)
\]

Substituting into (2):
\[
a + 10 + c = 13 \implies a + c = 3 \quad (5)
\]

Both (4) and (5) are consistent as they yield the same result. We now substitute \(b = 10\) into (3):

\[
9a + 3(10) + c = -7 \implies 9a + 30 + c = -7
\]

Rearranging this gives us:
\[
9a + c = -37 \quad (6)
\]

Now we can solve for \(a\) using equations (4) and (6). From equation (4), we know:
\[
c = 3 - a
\]

Substituting this into equation (6):
\[
9a + (3 - a) = -37 \implies 9a - a + 3 = -37 \implies 8a + 3 = -37 \implies 8a = -40 \implies a = -5
\]

Now substituting \(a = -5\) back to find \(c\):
\[
c = 3 - (-5) = 3 + 5 = 8
\]

So we have:
- \(a = -5\)
- \(b = 10\)
- \(c = 8\)

Thus, the equation of the parabola is:
\[
y = -5x^2 + 10x + 8
\]

This is the equation whose graph is a parabola that fits the points in the table.