graph-the-function-by-hand-not-by-plotting-points-but-by-starting-with-the-graph-of-one-of-the-st-27

Question

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section $1.2,$ and then applying the appropriate transformations.

$y=\frac{1}{2}(1-\cos x)$

Amrita Bhasin · Accepted Answer

this question asks us to graft the function by hand. We&#x27;re big given. Why is 1/2 times one minus co sign axe? The first thing we need to do is need to consider what our parent function is gonna be. We can consider our parent function to simply be co sign of acts without any shifts. Now, in order to get this from this, we need to shift that me to shrink the graph by a factor of two vertically, then reflect it with respect to the X axis. And then lastly, shift 1/2 is 1/2 over here, Units upwards. That&#x27;s a lot going on here. But let&#x27;s start with the parent function. Okay, Parent Function Co sign of acts. Looks like Bess given amplitude of one. We can just assume that So then all those things were gonna be drink shrinking by a factor of two vertically reflecting with respect to the X axis and shifting 1/2 unit upwards. It&#x27;s gonna look like that

Heather Zimmers · Answer

our coolest a graph dysfunction, but we&#x27;re going to start with the graph of one of the standard functions Y equals co Sign X and then figure out the transformations that took place. So the coastline graph has a period of two. Pi goes up to one and down to negative one and looks roughly like this. It goes on in both directions, but I&#x27;m just showing part of it. Okay, now what happens if we multiply it by two When we multiply it by two. That&#x27;s a vertical stretch, so now it&#x27;s going to be twice as tall. So now it&#x27;s going to go up to a height of two and down to a height of negative, too. We say the amplitude changed. Okay, so we can go ahead and sketch that twice is tall. All right, now what happens if we multiply that by negative one? Well, that&#x27;s going to reflect it across the X axis. Flip it upside down, you could say. So we&#x27;ll take what we just drew and make that move still going up to a height of one at a height of two down to a height of negative, too, but upside down to direct sketch. Okay? Finally, the graph that we&#x27;ve been waiting for Why equals three minus to co sign X. All we have to do now is at the three and adding the three is going to shift the whole graph of three. So now instead of the top point being at a height of two, the top point is going to be at her five a height of five. And instead of the bottom point being at a height of negative two, it&#x27;s going to be at a height of one. So let&#x27;s make room for that. So the middle line is going to be shifted up. Three. I&#x27;ll just draw in that middle line there to give us some reference. Okay, so we have this upside down graph going down to a height of one and up to a height of five. There we go.

Heather Zimmers · Answer

to graph this function, we&#x27;re going to start by looking at the graph of the standard function y equals sign of X, so we know that it has a period of two pi. That&#x27;s the with that one cycle, and we knew it goes up to a height of one and down to a height of negative one. So it looks roughly like this. Now what happens to it if we multiply the X by 1/2 that represents a horizontal stretch, so it&#x27;s going to be two times as wide. So we say horizontal stretch by a factor of two that changes the period. The width of one cycle is now going to be four pi instead of two pi. So if you want to draw it out that far, we can still goes up to a height of one and down to a height of negative one. That didn&#x27;t change. So now we have the graph that is stretched out wider, like so

Amrita Bhasin · Answer

this question asks us to Graff, why is sign of 1/2 X by hand? We know we&#x27;re gonna start with the parent function, which is half of axe is sign of acts and we know we get the given function. Why? By stretching the graph of f of X by a factor of two horizontally. So this, we know, is the normal sign function right now stretching the graph of f of X by a factor of two horizontally, you should get something like that.

Heather Zimmers · Answer

to graph dysfunction. Let&#x27;s start by thinking about the standard function Y equals co sign of X. What does that look like? Goes upto one down to negative one. It has a period of two pi. So the coast and graph in general looks like this. The regular standard Cho sang graph. Okay. And it keeps going forever to the left and the right. That&#x27;s just part of it. Now what happens if we multiply the X pipe? I What does that do that&#x27;s going to be a horizontal shrink by a factor of pie? That means it&#x27;s pi times narrower than it used to be roughly three times narrower. So the period, the width of one cycle used to be two pi. Now the period is going to be just too. We divide the period by the horizontal shrink factor, so the period is just going to be, too. That&#x27;s roughly three times narrower. Now, I&#x27;m not gonna make that perfectly to scale compared to the previous one. I&#x27;m just going to put a to here and a negative to here. We still go upto one down to negative one. That didn&#x27;t change. So roughly here is the graph we have now. But now the final thing we want to do is the absolute value signs. So the absolute value is going to make it positive. Anything that was previously positive will still be positive. Anything that was negative is going to now be reflected across the X axis and made positive. Okay, so I graft was positive for a while. Then it went negative. So that negative part is going to be reflected back up. And it was positive for a while, and then it went negative and that negative parts going to be reflected back up. And then it was positive for a while, So that&#x27;s what we get.

Heather Zimmers · Answer

our goal list. A graph y equals to co sign of three x So let&#x27;s start by thinking about why equals co sign of X to go from y equals co sign of X to y equals to co sign of three X. There&#x27;s going to be a vertical stretch by a factor of two, so two times is tall and there&#x27;s going to be a horizontal shrink by a factor of three, so three times narrower. Okay, let&#x27;s take this step by step. So let&#x27;s first start by thinking about how why equals co sign of X looks. It goes upto one down to negative one. It has a period of two pi. So the graph of Y equals co sign of X looks like this. Okay, now let&#x27;s take it one transformation at a time. Now suppose we want to look at why equals to co sign of X. So we&#x27;re taking into account the vertical stretch. So now, instead of going up to a height of one and down to a height of negative one, it goes up to a height of two and down to a height of negative, too. Still has a period of two pi. So that&#x27;s going to look like this same thing on the other side. Okay, Now let&#x27;s take into account the horizontal shrink, so that&#x27;s going to make it narrower and change the period. Gonna squeeze that one and down here. So instead of having a width of two pi for one complete cycle, we have a width of 1/3 of that. So two pi over three gives us one complete cycle, and I&#x27;m just gonna put the two pi over three right here. You would have three complete cycles in a width of two pi instead of one complete cycle in a width of two pi still going up to two and down to negative two. Okay, so one complete cycle in a width of two pi over three is going to look like that. And if you want to show that you have three times as many in a width of two pi, you can just do a whole bunch of those and you could do that on the other side as well

Problem

Graph the function by hand, not by plotting point…

Problem 17 Medium Difficulty

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section $1.2,$ and then applying the appropriate transformations.

$y=\frac{1}{2}(1-\cos x)$

Answer

blue graph represents $\mathrm{f}(\mathrm{x}),$ red graph represents $y=\frac{1}{2}-\frac{1}{2} \cos x$

( GRAPH NOT AVAILABLE )

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James Stewart

Biocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Kayleah T.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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blue graph represents $\mathrm{f}(\mathrm{x}),$ red graph represents $y=\frac{1}{2}-\frac{1}{2} \cos x$ ( GRAPH NOT AVAILABLE )

Biocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Kayleah T.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Kayleah T.

Joseph L.

blue graph represents $\mathrm{f}(\mathrm{x}),$ red graph represents $y=\frac{1}{2}-\frac{1}{2} \cos x$

( GRAPH NOT AVAILABLE )

James Stewart

Biocalculus Calculus for the Life Sciences