The parabola \(y = ax^2 + bx + c\) is graphed below. Find \(a+b+c\). (The grid lines are one unit apart.)
Using the graphing equation: y = a(x - h)2 + k where (h, k) is the vertex.
The vertex of this parabola occurs at (2, 1) ---> y = a(x - 2)2 + 1
It rises, so that the value of a is positive.
It has no stretching or contracting so the value of a is 1.
The equation is: y = 1(x - 2)2 + 1 or y = (x - 2)2 + 1.
Now, multiply this out and simplify ...
Notice that the curve goes through the following three points (you could choose any three, but these are easy ones):
(0, 5) (2, 1) and (4, 5)
This allows you to set up the following three equations:
5 = a*02 + b*0 + c or simply c = 5
1 = a*12 + b*1 + c or a + b = -4 ...(1)
5 = a*42 + b*4 + c or 16a + 4b = 0 ...(2)
You could solve equations (1) and (2) for a and b if you wish; however, since you are only asked for a + b + c you could simply use the value of a + b from (1) together with the value for c.