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Math Help - Finding vertex of parabola by completing the square?

  1. #1
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    Finding vertex of parabola by completing the square?

    I must have been absent from school when we learned this because the only way I know how to find the vertex of a parabola is to graph it and calculate the maximum. Help is appreciated. Thanks in advance.

    The equation of a parabola is y=x-2x+7. Show how to determine the vertex of this parabola by completing the square. What is the vertex of the parabola?
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  2. #2
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    Let's see how well I remember this one...

    Ok, so we reorganize the equation:

    x^2 - 2x - y = -7

    So we want to complete the squares for the x terms. Take the -2x term and divide by 2 (so, -1) and square that (thus, 1). That is the number you need to complete the square. Do this:

    x^2 - 2x + 1 - y = -7 + 1

    Note that I added the +1 to the right side as well!

    So now we can easily factor x^2 - 2x + 1, leaving us with:

    (x - 1)^2 - y = -6.

    Voila, square completed. You'll probably want to reorganize the equation a bit into the standard form:

    (x - 1)^2 = y - 6

    Can you take it from there?
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  3. #3
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    Thanks! I'm thinking the vertex is (1,-6). The x-value is negated because we're taking it our of parentheses and the y-value remains the same.
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  4. #4
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    Close!

    Actually, the vertex is at (1, 6). Remember that the formula is:

    (x - h)^2 = 4p(y - k), where the vertex is (h, k).

    In this case, 4p = 1, leaving just the (y - 6) on the right side for y = 6. Of course, with the (x - 1)^2, we get x = 1 for a vertex at (1, 6). Always good to graph it to make sure your solution is correct, I find.
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  5. #5
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    I see! Thank you once again!
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  6. #6
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    My pleasure, glad to help.
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