Find the equation of the parabola passing through the given points. $( - 2,3 ) , ( 1,12 )$, and $( 2,19 )$
A) $\quad y = 4 x + 6$
B) $\quad y = x ^ { 2 } - 4 x + 7$
C) $\quad y = x ^ { 2 } + 4 x + 7$
D) $\quad y = - 5 x ^ { 2 } + 4 x + 31$
E) none of these

Question

Quizplus · Accepted Answer

We can use the general equation of a parabola:

$$y = ax^2 + bx + c$$

Substituting the given points, we get a system of three equations:

$$3 = 4a - 2b + c$$
$$12 = a + b + c$$
$$19 = 4a + 2b + c$$

Solving for $a$, $b$, and $c$ using elimination or substitution, we get:

$$a = 1$$
$$b = 4$$
$$c = 7$$

Substituting in the general equation, we get:

$$\boxed{y = x^2 + 4x + 7}$$

Find the Equation of the Parabola Passing Through the Given (−2,3),(1,12)( - 2,3 ) , ( 1,12 )(−2,3),(1,12)

Find the Equation of the Parabola Passing Through the Given $( - 2,3 ) , ( 1,12 )$