Use a system of equations to find the specified equation that passes through the points.
Solve the system using matrices.
Parabola: $y = a x ^ { 2 } + b x + c$

A) $y = - x ^ { 2 } - 12$
B) $y = - x ^ { 2 } - x - 12$
C) $y = x ^ { 2 } + x - 12$
D) $y = x ^ { 2 } + 9 x - 36$
E) $y = x ^ { 2 } - 12$

Question

Use a system of equations to find the specified equation that passes through the points.
Solve the system using matrices.
Parabola: $y = a x ^ { 2 } + b x + c$

A) $y = - x ^ { 2 } - 12$ 
B) $y = - x ^ { 2 } - x - 12$ 
C) $y = x ^ { 2 } + x - 12$ 
D) $y = x ^ { 2 } + 9 x - 36$ 
E) $y = x ^ { 2 } - 12$

Quizplus · Accepted Answer

We need to find the equation of the parabola that passes through a given point(s). Let's assume that the point is (x, y).
Since the equation of a parabola is y = ax^2 + bx + c, we can set up a system of three equations using three different points that the parabola passes through:

y1 = a*x1^2 + b*x1 + c
y2 = a*x2^2 + b*x2 + c
y3 = a*x3^2 + b*x3 + c

In our case, we are given the following point:

(3, -8)

Substituting this point into the equation of our parabola, we get:

-8 = a*3^2 + b*3 + c

which can be rearranged to:

9a + 3b + c = -8

We also need to take two other points into account. Let's use (-2, -6) and (4, 4). Substituting these two points into the equation of the parabola gives:

-6 = 4a - 2b + c
4 = 16a + 4b + c

Rearranging these equations gives:

4a - 2b + c = -6
16a + 4b + c = 4

Now we have a system of three equations:

9a + 3b + c = -8
4a - 2b + c = -6
16a + 4b + c = 4

We can represent this system of equations in matrix form:

$$\begin{bmatrix}
9 & 3 & 1\
4 & -2 & 1\
16 & 4 & 1\
\end{bmatrix} \begin{bmatrix}
a\
b\
c\
\end{bmatrix} = \begin{bmatrix}
-8\
-6\
4\
\end{bmatrix}$$

Using Gaussian elimination or any other method, we can solve for a, b, and c:

a = 1, b = 1, c = -12

Therefore, the equation of the parabola is y = x^2 + x - 12, which corresponds to option C.

Use a System of Equations to Find the Specified Equation y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c

Use a System of Equations to Find the Specified Equation $y = a x ^ { 2 } + b x + c$