Question

Write the equation of the parabola that passes through the points (−3,−4)
, (−4,0)
, and (2,6)
.(1 point)
Responses

y=x2+3x−4
y equals x squared plus 3 x minus 4

y=−x2−x+12
y equals negative x squared minus x plus 12

y=−6x2−11x+52
y equals negative 6 x squared minus 11 x plus 52

y=6x2+13x−44

Answer 1

To find the equation of the parabola passing through the points \((-3, -4)\), \((-4, 0)\), and \((2, 6)\), we can use the general form of a quadratic equation:

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

We will substitute each of the given points into this equation to create a system of equations.

1. **Using \((-3, -4)\)**:
\[
-4 = a(-3)^2 + b(-3) + c \\
-4 = 9a - 3b + c \quad \text{(Equation 1)}
\]

2. **Using \((-4, 0)\)**:
\[
0 = a(-4)^2 + b(-4) + c \\
0 = 16a - 4b + c \quad \text{(Equation 2)}
\]

3. **Using \((2, 6)\)**:
\[
6 = a(2)^2 + b(2) + c \\
6 = 4a + 2b + c \quad \text{(Equation 3)}
\]

Now we have a system of equations:
1. \( 9a - 3b + c = -4 \)
2. \( 16a - 4b + c = 0 \)
3. \( 4a + 2b + c = 6 \)

Next, we can eliminate \(c\) by subtracting one equation from another.

**Subtract Equation 1 from Equation 2**:
\[
(16a - 4b + c) - (9a - 3b + c) = 0 - (-4) \\
16a - 9a - 4b + 3b = 4 \\
7a - b = 4 \quad \text{(Equation 4)}
\]

**Subtract Equation 1 from Equation 3**:
\[
(4a + 2b + c) - (9a - 3b + c) = 6 - (-4) \\
4a + 2b - 9a + 3b = 10 \\
-5a + 5b = 10 \\
-a + b = 2 \quad \text{(Equation 5)}
\]

Now we have two new equations:
1. \( 7a - b = 4 \) (Equation 4)
2. \( -a + b = 2 \) (Equation 5)

Now we will solve this system. From Equation 5, we can express \(b\) in terms of \(a\):
\[
b = a + 2
\]

Now substitute \(b\) into Equation 4:
\[
7a - (a + 2) = 4 \\
7a - a - 2 = 4 \\
6a - 2 = 4 \\
6a = 6 \\
a = 1
\]

Now substituting \(a = 1\) back into Equation 5:
\[
-b = 1 + 2 \\
b = 3
\]

Now we have \(a = 1\) and \(b = 3\). We need to find \(c\) using one of our original equations. Let's use Equation 1:
\[
9(1) - 3(3) + c = -4 \\
9 - 9 + c = -4 \\
c = -4
\]

So the coefficients are:
\[
a = 1, \quad b = 3, \quad c = -4
\]

The equation of the parabola is:
\[
y = 1x^2 + 3x - 4
\]

Which simplifies to:
\[
y = x^2 + 3x - 4
\]

Thus, the equation of the parabola that passes through the points \((-3, -4)\), \((-4, 0)\), and \((2, 6)\) is:
\[
\boxed{y = x^2 + 3x - 4}
\]