Provide an appropriate response.
-The curve $y = a x ^ { 2 } + b x + c$ passes through the point $( 2,12 )$ and is tangent to the line $y = 4 x$ at the origin. Find a, $b$, and $c$.
A) a = 2, b = 0, c = 0
B) a = 0, b = 1, c = 4
C) a = 1, b = 4, c = 0
D) a = 4, b = 0, c = 1

Question

Quizplus · Accepted Answer

Since the curve is tangent to the line at the origin, we know that the equation of the curve at the origin is also $y = 4x$. Therefore, we have the system of equations:
$$a(0)^2 + b(0) + c = 0$$
$$a(2)^2 + b(2) + c = 12$$
$$a = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 0}{2 - 0} = 6$$
Solving for $b$ and $c$, we get $b = 4a$ and $c = -4a$. Substituting $a = 6$ gives us $b = 24$ and $c = -24$, so the answer is C.

Provide an Appropriate Response y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c

Provide an Appropriate Response $y = a x ^ { 2 } + b x + c$