Find $a+b+c$ if the graph of the equation $y=ax^2+bx+c$ is a parabola with vertex $(5,3)$, vertical axis of symmetry, and contains the point $(2,0)$.
+128408 Find $a+b+c$ if the graph of the equation $y=ax^2+bx+c$ is a parabola with vertex $(5,3)$, vertical axis of symmetry, and contains the point $(2,0)$.
We have the vertex form
y = a(x - h)^2 + k where (h, k) = ( 5, 3)
And since the point (2,0) is on the graph, we can slove for "a" thusly
0 = a(2 - 5)^2 + 3 subract 3 from both sides
-3 = a(-3)^2
-3 = 9a divide both sides by 9
(-1/3) = a
So we have
y = (-1/3) (x - 5)^2 + 3
y = (-1/3) [ x^2 - 10x + 25) + 3
y =(-1/3)x^2 + (10/3)x - 25/3 + 9/3
y = (-1/3)x^2 + (10/3)x - 16/3
So
a + b + c =
(-1/3) + (10/3) + (-16/3) =
- 7 / 3
