Question

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

y
=
−
x
2
−
x
+
12
y equals negative x squared minus x plus 12

y
=
−
6
x
2
−
11
x
+
52
y equals negative 6 x squared minus 11 x plus 52

y
=
6
x
2
+
13
x
−
44
y equals 6 x squared plus 13 x minus 44

y
=
x
2
+
3
x
−
4

Answer 1

To find the equation of the parabola that passes 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 need to determine the coefficients \(a\), \(b\), and \(c\) using the given points.

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

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

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

Now we have a system of three equations:

1. \(9a - 3b + c = -4\)
2. \(16a - 4b + c = 0\)
3. \(4a + 2b + c = 6\)

Next, we'll eliminate \(c\) from the equations.

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

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

Now, we can solve the equations \(7a - b = 4\) (Equation 4) and \(-a + b = 2\) (Equation 5).

From Equation 5, express \(b\) in terms of \(a\):
\[
b = a + 2
\]

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

Now substituting \(a = 1\) back into the equation for \(b\):
\[
b = 1 + 2 = 3
\]

Now we substitute \(a\) and \(b\) back into one of the original equations to find \(c\). Using Equation 1:
\[
9(1) - 3(3) + c = -4
\]
\[
9 - 9 + c = -4 \implies c = -4
\]

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

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

This matches one of the given options:
\[
\boxed{y = x^2 + 3x - 4}
\]