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We're looking at transformations here in this problem and for
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each one. Let's take a look at the transformation
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in general and then apply it to the problem.
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So in general, if you want to shift a
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graph up, if you start with Y equals F
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of X, it will change to why equals F
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of X plus C, and that shifts it up
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. See units. So if we want to shift
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our graph up three, it's going to change to
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y equals F of X plus three. Similarly,
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if you want to shift a graph down in general
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, if you start with Y equals F of X
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, it's going to change to y equals F of
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X minus C, and that will shift it down
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, see units. So for our specific example,
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ours would change to y equals F of X minus
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three. Now let's look it right and left.
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So in general, if you want to shift something
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to the right and you start with y equals F
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of X, it's going to change to why equals
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f of X minus C, and that's going to
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shift it si units to the right. That's counterintuitive
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. Most of us expected to be X plus C
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. But it's the opposite of what we expect.
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So for the example we have here, the problem
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we have here if we're shifting at right three,
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we're going to end up with y equals F of
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X minus three and then just the opposite for left
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. If we start in general with y equals F
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of X and we shipped it left si units,
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it will look like Michael's F of X plus C
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. So for our problem, we're shifting. It
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left three units, so it's going to look like
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why equals F of X plus three. Now it's
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Think about reflections. An X axis reflection is actually
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a vertical reflection. It goes from being in se
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quarter one to quarter four or from being saying quadrant
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two to quadrant three. So anything that's positive becomes
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negative. Anything that's negative becomes positive. So that's
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reflected this way in general, if you start with
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why equals F of X and you reflected about the
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X axis you get, why equals the opposite of
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F of X? Positive Y values become negative.
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Negative y values become positive. So for this specific
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problem, then, ah, there's nothing more we
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can do except state that again, Why equals the
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opposite of F of X? There's no number to
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plug in here. Same idea for a Y axis
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reflection. That's a horizontal reflection. If you start
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with y equals F of X, it's going to
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change to y equals f of the opposite of X
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. All the positive X values changed too negative,
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and all the negative X values changed a positive,
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so there's no specific number to plug in here for
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that either. And then finally, vertical stretches and
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vertical shrinks. So in general, if you want
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to stretch a graph vertically, you're going to have
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to multiply it by a number greater than one.
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So y equals f of X becomes Why equals see
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Time's F of X, where C is greater than
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one. So for this specific problem, we would
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have why equals three times F of X and then
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for a vertical shrink. In general, we're going
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to start with y equals f of X, and
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we're going to multiply it by a number that's between
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zero and one. We can still call it see
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, so zero is less than C is less than
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one and so here. If it's a vertical shrink
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by a factor of three, that means it's going
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to be three times shorter, so we're multiplying it
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by 1/3 so we get y equals 1/3 times f
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of x.