Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( 0,2 ) , ( 1,2 ) , ( 2,4 )$
A) $y = x ^ { 2 } + x - 2$
B) $y = x ^ { 2 } - x - 2$
C) $y = 2 x ^ { 2 } - x + 2$
D) $y = 2 x ^ { 2 } - x + 1$
E) $y = x ^ { 2 } - x + 2$

Question

Quizplus · Accepted Answer

To find the equation of the parabola, we substitute the given points into the general equation $y = ax^2 + bx + c$ and solve for $a$, $b$, and $c$.1. Substituting $(0,2)$ gives $2 = a(0)^2 + b(0) + c$, so $c = 2$.2. Substituting $(1,2)$ gives $2 = a(1)^2 + b(1) + 2$, leading to $a + b = 0$.3. Substituting $(2,4)$ gives $4 = a(2)^2 + b(2) + 2$, which simplifies to $4a + 2b = 2$.Solving the system of equations $a + b = 0$ and $4a + 2b = 2$ (after substituting $c = 2$) gives $a = 1$ and $b = -1$. Therefore, the equation of the parabola is $y = x^2 - x + 2$, which matches choice 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 $y = a x ^ { 2 } + b x + c$