Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. ?? $( 0,5 ) , ( 1,6 ) , ( 2,5 )$ ?
A)? $x ^ { 2 } + 2 x + 5$
B)? $- x ^ { 2 } - 2 x - 5$
C)? $- x ^ { 2 } - 2 x + 5$
D)? $- x ^ { 2 } + 2 x + 5$
E)? $- x ^ { 2 } + 2 x - 5$

Question

Quizplus · Accepted Answer

Since a parabola can be uniquely defined by three non-collinear points, we can use the given two points along with the vertex, which is the midpoint of the line segment connecting the two points, to find the equation of the parabola. The midpoint of (1,4) and (5,-2) is ((1+5)/2, (4-2)/2) = (3,1). This is the vertex of the parabola, which means the equation has the form y = a(x-3)^2 + 1 for some value of a. To find a, we use the fact that the point (7,1) lies on the parabola:
1 = a(7-3)^2 + 1
0 = 16a
a = 0
Therefore, the equation of the parabola is y = 1, which is represented by answer choice D.

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 $y = a x ^ { 2 } + b x + c$