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Problem 5 Medium Difficulty

The graph of $ f $ is given. Use it to graph the following functions.

(a) $ y = f (2x) $
(b) $ y = f (\frac{1}{2} x) $
(c) $ y = f (-x) $
(d) $ y = -f (-x) $

Answer

(a) To graph $y=f(2 x)$ we shrink the graph of $f$ horizontally by a factor of 2.
The point (4,-1) on the graph of $f$ corresponds to the point $\left(\frac{1}{2} \cdot 4,-1\right)=(2,-1)$
(b) To graph $y=f\left(\frac{1}{2} x\right)$ we stretch the graph of $f$ horizontally by a factor of 2 .
The point (4,-1) on the graph of $f$ corresponds to the point $(2 \cdot 4,-1)=(8,-1)$.
(c) To graph $y=f(-x)$ we reflect the graph of $f$ about the $y$ -axis.
The point (4,-1) on the graph of $f$ corresponds to the \[ \text { point }(-1 \cdot 4,-1)=(-4,-1)\].
(d) To graph $y=-f(-x)$ we reflect the graph of $f$ about the $y$ -axis, then about the $x$ -axis.
The point (4,-1) on the graph of $f$ corresponds to the \[\text { point }(-1 \cdot 4,-1 \cdot-1)=(-4,1)\].

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RJ

Raghad J.

July 22, 2021

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Video Transcript

all right, we have the graph of F and we're going to graph several transformations. Starting with y equals F of two X, So that's going to be a horizontal shrink by a factor of two. So every point is going to have half the X coordinate that it had before. So a point with an ex coordinated zero is still a zero a point with the next coordinative one is now at 1/2 a point with the next coordinated four is now at two, and a point with an X coordinate of six is now at three. Notice that the height did not change it all, just the width. Next in part B, we see we're changing it to y equals F of 1/2 X, and that's going to be a horizontal stretch by a factor of two. So the graph is going to be two times this white, so we can take the points and double their X coordinates. So point with an ex coordinated zero is still zero a point with the next coordinative. One is now a two. A point with an X coordinate of four is now eight and a point with an X coordinate of six is now a 12. So then we can connect those points. Notice that the graph did not change in its height. Didn't get taller or shorter. It just got twice as wide. And this next transformation is a reflection across the Y axis. All the positive X values air going to become negative. And if you have any negative X values, they would become positive. So he have an X value of zero is still zero. The next value of one is now negative one, an X value of four is now negative for and an X value of six is now negative. Six. And if we connect those points, we get our new graph, which looks like a reflection of our old one across the y axis. And for this transformation, we see that X axis reflection that we saw in the last problem. But we also see a Y axis reflection, so we have X axis and y axis reflections together. Those make what we call an origin reflection, and it's going to look like the graph is turned 180 degrees so we can start by doing the X axis reflection and then do the Y axis reflection. So the X axis reflection would take us down here. And then the final graph that were interested in will then have the Y axis reflection. So just concentrate on the green one. Ignore the red one and the green one is the answer.