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Math Help - More problems with analytical geometry

  1. #1
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    More problems with analytical geometry

    Hello

    I have a problem with translating a parabola after a series of other transformations.
    Given the parabola y= (x-2)(x-5) find the turning points. I have done this ( 3.5, -2.25).
    Reflect this about a horizontal line passing through its turning point. I have also done this, with the equation now being y= (-x^2 -4.5)+7x -10

    1.The function is now translated 4 units to the left.

    At this point I am lost as to where I insert the next translations into the equation.

    I can hand graph this onto paper but the equation escapes me.

    2. The function is now stretched by a factor of 2.

    I am also lost at this point.

    Any assistance would be appreciated.

    Thank you.
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  2. #2
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    Quote Originally Posted by richie View Post
    Hello

    I have a problem with translating a parabola after a series of other transformations.
    Given the parabola y= (x-2)(x-5) find the turning points. I have done this ( 3.5, -2.25).
    Reflect this about a horizontal line passing through its turning point. I have also done this, with the equation now being y= (-x^2 -4.5)+7x -10

    1.The function is now translated 4 units to the left.

    At this point I am lost as to where I insert the next translations into the equation.

    I can hand graph this onto paper but the equation escapes me.

    2. The function is now stretched by a factor of 2.

    I am also lost at this point.

    Any assistance would be appreciated.

    Thank you.
    Transforming a parabola is always easiest if you put it in general form

    y = a(x - h)^2 + k

    where a is the dilation factor, (h, k) is the turning point, and also represents the translations.


    You have

    y = (x - 2)(x - 5)

     = x^2 - 5x - 2x + 10

     = x^2 - 7x + 10

     = x^2 - 7x + \left(-\frac{7}{2}\right)^2 - \left(-\frac{7}{2}\right)^2 + 10

     = \left(x - \frac{7}{2}\right)^2 - \frac{9}{4}.


    So the turning point is (x, y) = \left(\frac{7}{2}, -\frac{9}{4}\right).


    To reflect this, you need to negate the dilation factor, while keeping the turning point the same.

    So if the function is reflected about the turning point

    y = -\left(x - \frac{7}{2}\right)^2 - \frac{9}{4}.


    To translate the function 4 units to the left, that means that the turning point will move to \frac{7}{2} - 4 = -\frac{1}{2}.

    So your function now becomes

    y = -\left(x + \frac{1}{2}\right)^2 - \frac{9}{4}.


    Finally, to STRETCH a function by a factor, divide your dilation factor by that factor.

    So since you're stretching by a factor of 2, that means you divide your dilation factor by 2.


    The function finally becomes

    y = -\frac{1}{2}\left(x + \frac{1}{2}\right)^2 - \frac{9}{4}.
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  3. #3
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    more problems with analytical geometry

    Dear Prove It

    Thanks for your help and assistance I appreciate it.

    Thanks.
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