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Math Help - Graph

  1. #1
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    Graph

    Graph y = sqrt{x^2}, y = x, y |x|, and y = (sqrt{x})^2, NOTING which graphs are the SAME.

    NOTE:

    What is the mirror-like image of the graph y = x?
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  2. #2
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    Quote Originally Posted by symmetry View Post
    Graph y = sqrt{x^2}, y = x, y |x|, and y = (sqrt{x})^2, NOTING which graphs are the SAME.

    NOTE:

    What is the mirror-like image of the graph y = x?
    y = (sqrt(x))^2 is the same as --> y = x; y = sqrt(x^2) --> y = abs(x)

    The mirror image is y = -x
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  3. #3
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    Quote Originally Posted by symmetry View Post
    Graph y = sqrt{x^2}, y = x, y |x|, and y = (sqrt{x})^2, NOTING which graphs are the SAME.

    NOTE:

    What is the mirror-like image of the graph y = x?
    I'm not going to graph these, you should be able to do that yourself. However I will note that only two of these graphs are the same. The trick is in the last graph: \sqrt{x^2} and (\sqrt{x})^2 are "apparently" the same since each reduces to x, but note that in the first expression the domain is  (-\infty, \infty ) and in the last second expression is  [0, \infty ).

    y = \sqrt{x^2} and y = x aren't the same either since \sqrt{.} only returns a positive value. It is for this very reason that y = \sqrt{x^2} and y = |x| have the same graph.

    Quote Originally Posted by symmetry View Post
    What is the mirror-like image of the graph y = x?
    Depends on which "mirror" you are talking about. The typical mirror "planes" are the x and y axes, and the lines y = x and y = -x.

    y = x reflected over the x-axis is y = -x.
    y = x reflected over the y-axis is y = -x.
    y = x reflected over the line y = x is y = x.
    y = x reflected over the line y = -x is y = x.

    -Dan
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by AfterShock View Post
    y = (sqrt(x))^2 is the same as --> y = x; y = sqrt(x^2) --> y = abs(x)

    The mirror image is y = -x
    I've posted a graph of both functions. y = \sqrt{x^2} is in solid red and y = x is the dotted blue. It's hard to see the overlap in the first quadrant, but you know it's there.

    -Dan
    Attached Thumbnails Attached Thumbnails Graph-y-equals-x.jpg  
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    Quote Originally Posted by topsquark View Post
    I've posted a graph of both functions. y = \sqrt{x^2} is in solid red and y = x is the dotted blue. It's hard to see the overlap in the first quadrant, but you know it's there.

    -Dan
    Indeed; as far as I could tell, he was asking for the mirror image of just y = x, not the original reflected over y = x.

    Wasn't very clear.
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by AfterShock View Post
    Indeed; as far as I could tell, he was asking for the mirror image of just y = x, not the original reflected over y = x.

    Wasn't very clear.
    Actually I wasn't very clear, sorry. I was responding to your comment that y = (\sqrt{x})^2 and y = x have the same graph. But I graphed the wrong function. These aren't the same because the first only has the domain  [0, \infty ), whereas the second has a domain of all real numbers.

    -Dan
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