SOLUTION: A quadratic function has zeros 3 and 9. It passes through point (6, 11). What is the vertex of the graph?

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Question 1179958: A quadratic function has zeros 3 and 9. It passes through point (6, 11).
What is the vertex of the graph?
Found 3 solutions by josgarithmetic, MathLover1, greenestamps:
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
y=a%28x-3%29%28x-9%29, based on given zeros

a%286-3%29%286-9%29=11, based on the included point
a%2A3%2A%28-3%29=11
a=-11%2F9

y=-%2811%2F9%29%28x-3%29%28x-9%29

The x value for vertex will occur in the exact middle of the two zeros.
%283%2B9%29%2F2=6----------find y for x=6.
That x y pair will be the vertex point.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=a%28x-x%5B1%5D%29%28x-x%5B2%5D%29->has zeros x%5B1%5D=3+and x%5B2%5D=9
f%28x%29=a%28x-3%29%28x-9%29.....passes through point (6, 11)
11=a%286-3%29%286-9%29
11=a%283%29%28-3%29
11=-9a
a=-11%2F9

f%28x%29=-%2811%2F9%29%28x-3%29%28x-9%29........write it in vertex form
f%28x%29=+-%2811%2F9%29+%28x%5E2+-+12x+%2B+27%29...........complete square
f%28x%29=+-%2811%2F9%29+%28%28x%5E2+-+12x%2Bb%5E2%29-b%5E2+%2B+27%29
f%28x%29=+-%2811%2F9%29+%28%28x%5E2+-+12x%2B6%5E2%29-6%5E2+%2B+27%29
f%28x%29=+-%2811%2F9%29+%28%28x+-+6%29%5E2-36+%2B+27%29
f%28x%29=+-%2811%2F9%29+%28%28x+-+6%29%5E2-9%29...take -9 out
f%28x%29=+-%2811%2F9%29+%28%28x+-+6%29%5E2%29-%2811%2F9%29%28-9%29
f%28x%29=+-%2811%2F9%29+%28%28x+-+6%29%5E2%29%2B11
the vertex of the graph is at (6,11)






Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


You have two answers that both use valid algebraic techniques to find the answer.

You should understand those algebraic methods. However, you should know that all the work they did to answer this question was a waste of time.

The zeros of the function are 3 and 9, so the axis if symmetry is x=6. So if the graph passes through (6,11), then (6,11) is the vertex.

ANSWER: (in a few seconds, without algebra) The vertex is (6,11).