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Problem 1 Medium Difficulty

Suppose the graph of $ f $ is given. Write equations for the graphs that are obtained from the graph of
$ f $ as follows.

(a) Shift 3 units upward.
(b) Shift 3 units downward.
(c) Shift 3 units to the right.
(d) Shift 3 units to the left.
(e) Reflect about the x-axis.
(f) Reflect about the y-axis.
(g) Stretch vertically by a factor of 3.
(h) Shrink vertically by a factor of 3.

Answer

(a) If the graph of $f$ is shifted 3 units upward, its equation becomes $y=f(x)+3$.
(b) If the graph of $f$ is shifted 3 units downward, its equation becomes $y=f(x)-3$.
(c) If the graph of $f$ is shifted 3 units to the right, its equation becomes $y=f(x-3)$.
(d) If the graph of $f$ is shifted 3 units to the left, its equation becomes $y=f(x+3)$.
(e) If the graph of $f$ is reflected about the $x$ -axis, its equation becomes $y=-f(x)$.
(f) If the graph of $f$ is reflected about the $y$ -axis, its equation becomes $y=f(-x)$.
(g) If the graph of $f$ is stretched vertically by a factor of 3 , its equation becomes $y=3 f(x)$.
(h) If the graph of $f$ is shrunk vertically by a factor of $3,$ its equation becomes $y=\frac{1}{3} f(x)$.

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Video Transcript

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

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