WEBVTT
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The figure shows the graph of π of π₯.
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A transformation maps π of π₯ to two π of π₯.
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Determine the coordinates of π΄ following this transformation.
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Now, weβre going to look at this question in two ways.
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The first method is to think about the algebraic representation of each of our transformations.
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Specifically, suppose we have the graph of a function π¦ equals π of π₯.
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This is mapped onto the graph of π¦ equals π π of π₯ for some constant π by a vertical stretch scale factor π.
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So, for the transformation that maps π of π₯ to two π of π₯, weβre performing a vertical stretch by a scale factor of two.
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Now, unfortunately, we canβt fully draw this on the diagram given, but we can assume that it might look a little something like this.
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All of the π¦-values in the coordinates are doubled.
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This means it does intersect the π₯-axis at the exact same points.
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Now, specifically, weβre interested in coordinate π΄, which is 180, negative one.
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We said that the π₯-values remain the same, but that the π¦-values are doubled.
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So, the image of π΄, the coordinates of π΄, after its transformation, which weβll call π΄ prime, has a π¦-value of negative one times two.
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So π΄ prime is 180, negative one times two or 180, negative two.
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So weβve shown an algebraic interpretation of this transformation.
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But we said there was another way.
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Now, that way is to think about the actual equation of the original graph.
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We have that sinusoidal shape.
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It has a π¦-intercept of one, maxima and minima at one and negative one, respectively, and it appears to repeat, be periodic, with a period of 360.
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We can therefore say that π of π₯ must be cos of π₯.
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Now, in fact, we can check this by substituting π₯ equals 180 into this function and check in we get negative one out.
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Well, π of 180 is cos of 180, which is in fact negative one.
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So, weβve identified the equation of π of π₯.
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Itβs cos of π₯.
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This means the function that weβre interested in after the transformation must be two π of π₯, and thatβs two cos of π₯.
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We could then plot the graph of π¦ equals two cos of π₯ on our axes.
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We could do so using a table or any other suitable method.
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Either way, we once again see that we can map π of π₯ under two π of π₯ by a vertical stretch with a scale factor of two.
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Either way, we find that the coordinates of π΄ following our transformation is 180, negative two.