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Thread: transfromations of an exponential function??

  1. #1
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    Question transfromations of an exponential function??

    Consider the graph of y = ex. (a) Find the equation of the graph that
    results from reflecting about the line y = 8.

    y = 1
    _________________________

    (b) Find the equation of the graph that results from reflecting about
    the line x = 2.

    y =_____________________________

    2
    How would i solve this, and can u show the steps?
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  2. #2
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    Hello Sneaky
    Quote Originally Posted by Sneaky View Post
    Consider the graph of y = ex. (a) Find the equation of the graph that
    results from reflecting about the line y = 8.
    Suppose that the point $\displaystyle (x_1, y_1)$ is transformed into the point $\displaystyle (x_2, y_2)$, by a reflection in the line $\displaystyle y = 8$. Then, since the reflection line is horizontal, the x-coordinate doesn't change, and so:

    $\displaystyle x_2 = x_1$ (1)

    The fundamental property of a reflection is that the distances of a point and its reflection from the mirror-line are equal. Let's assume (and it won't make any difference if it's the other way around) that $\displaystyle y_1 < 8$ and $\displaystyle y_2 > 8$. Then these distances are $\displaystyle (8-y_1)$ and $\displaystyle (y_2-8)$, respectively. So

    $\displaystyle 8-y_1 = y_2-8$

    $\displaystyle \Rightarrow y_2 = 16-y_1$ (2)

    So, since $\displaystyle y_1 = e^{x_1}$, the equation relating $\displaystyle y_2$ and $\displaystyle x_2$, when we use (1) and (2), is:

    $\displaystyle y_2 = 16 - e^{x_2}$

    and so the equation of the reflected graph is

    $\displaystyle y = 16-e^x$


    (b) Find the equation of the graph that results from reflecting about
    the line x = 2.

    y =_____________________________

    2
    How would i solve this, and can u show the steps?
    Do this in the same way. Assume that $\displaystyle (x_1,y_1)\rightarrow (x_2,y_2)$. Then, since the mirror-line is vertical

    $\displaystyle y_2=y_1$

    and this time it's the horizontal distances from the mirror-line that are equal. So

    $\displaystyle 2-x_1 = x_2-2$

    $\displaystyle \Rightarrow x_1 = 4-x_2$

    And, once again, $\displaystyle y_1=e^{x_1}$, so how are $\displaystyle y_2$ and $\displaystyle x_2$ related? This will then give you the equation of the transformed graph.

    Can you complete this now?

    Grandad
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