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# Parabola

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The graph of y=ax^2+bx+c is given below, where a, b, and c are integers. Find a-b+c.

Apr 15, 2022

#1
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Vertex form    (3,1)

y = a ( x -3)^2 +1       point 0,7      shows a =     2/3

y = 2/3 (x-3)^2 +1        now expand the right side to find a b and c

Apr 15, 2022
#2
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We can graph the parabola using vertex form: $$y = a(x-h)+k$$

We know the vertex is $$(3,1)$$. Plugging these values in for h and k, we get: $$y=a(x-3)^2+1$$

To solve for a, we have to plug in coordinates. In this case, I will use $$(0, 7)$$, but you will get the same answer with a different point.

We now have: $$7=a(-3)^2+1$$

Solving, we find $$a = {2 \over 3}$$

Now, we have: $$y = {2 \over 3} (x-3)^2+1$$

Expanding the right-hand side, we get: $$y = -{2\over3}x^2-4x+7$$

Now, we have: $$-{2 \over 3} - (-4)+7 = \color{brown}\boxed{10 {1 \over3}}$$

Apr 15, 2022