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) $$y = x ^ { 2 } - 3 x + 4$$
B) $$y = - x ^ { 2 } - 3 x + 1$$
C) $$y = - x ^ { 2 } + 3 x + 4$$
D) $$y = - x ^ { 2 } + 3 x + 1$$
E) $$y = - x ^ { 2 } - 2 x + 1$$

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) $$y = x ^ { 2 } - 3 x + 4$$ 
B) $$y = - x ^ { 2 } - 3 x + 1$$ 
C) $$y = - x ^ { 2 } + 3 x + 4$$ 
D) $$y = - x ^ { 2 } + 3 x + 1$$ 
E) $$y = - x ^ { 2 } - 2 x + 1$$

Quizplus · Accepted Answer

We need to find the equation of a parabola that passes through a given point, which means we have three variables (a, b, and c) to solve for. Hence, we need three equations. 
Substituting the given point into the equation of the parabola yields:
$$y = a x^2 + b x + c$$
$$3 = a(1)^2 + b(1) + c$$
$$4 = a(2)^2 + b(2) + c$$
$$1 = a(-1)^2 + b(-1) + c$$
Simplifying the equations give us:
$$a + b + c = 3$$
$$4a + 2b + c = 4$$
$$a - b + c = 1$$
Writing this as a matrix equation gives:
$$\begin{bmatrix}1&1&1\4&2&1\1&-1&1\end{bmatrix}\begin{bmatrix}a\b\c\end{bmatrix} = \begin{bmatrix}3\4\1\end{bmatrix}$$
Using Gaussian elimination or other methods, we can solve for a, b, and c. The resulting equation is:
$$y = -x^2 + 3x + 4$$
which matches choice 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 $y = a x ^ { 2 } + b x + c$