Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points.
$$( - 3,9 ) , ( - 2,7 ) , ( - 1,7 )$$
A) $y = x ^ { 2 } - 3 x - 9$
B) $y = x ^ { 2 } + 3 x - 9$
C) $y = 2 x ^ { 2 } + 3 x + 0$
D) $y = 2 x ^ { 2 } + 3 x + 5$
E) $y = x ^ { 2 } + 3 x + 9$

Question

Quizplus · Accepted Answer

To find the equation of the parabola, we substitute each point into the general equation $y = ax^2 + bx + c$ and solve for $a$, $b$, and $c$.For $(-3, 9)$:$$9 = a(-3)^2 + b(-3) + c$$$$9 = 9a - 3b + c$$ (1)For $(-2, 7)$:$$7 = a(-2)^2 + b(-2) + c$$$$7 = 4a - 2b + c$$ (2)For $(-1, 7)$:$$7 = a(-1)^2 + b(-1) + c$$$$7 = a - b + c$$ (3)Solving these equations simultaneously for $a$, $b$, and $c$ gives us the values that fit all three points. After solving, we find that $a = 1$, $b = 3$, and $c = 9$, which corresponds to the equation $y = x^2 + 3x + 9$. Therefore, the correct choice is E.

Find the Equation of the Parabola y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c

Find the Equation of the Parabola $y = a x ^ { 2 } + b x + c$