In Exercises 79–82, graph f, g, and h in the same rectangular ...

Question

In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)

Accepted Answer

Welcome back everyone. In this problem, we want to plot the functions of FG and H in the same coordinate system for X between zero and two pi by adding or subtracting the Y coordinates for our functions F FX equals four multiplied by the cosine of two xyeg FX, sorry equals three multiplied by the cosine of X and H FX equals F plus G of X. Now, what do we already know? Well, we want to put them in the same coordinate system. That means we're drawing them on the same graph and we already have values for F FX which is four multiplied by the cosine of two XG FX which is three multiplied by the cosine of X. But we'll have to figure out what, what H FX is. Well, recall that F plus G FX is the same thing as F FX plus GF. So we'll be adding F and G. So that tells us then that H FX would be F, which is four multiplied by the cosine of two X plus G FX, which is three multiplied by the cosine of X. Now, we know that F FX and G FX are cosine graphs. And we also know that for the cosine graph, generally speaking for Y equals a multiplied by the cosine of BX minus C plus D. That's the general form for a trigonometric graph. Then the amplitude of our graph equals A and the period for our graph equals two P, two pi divided by B. And by looking on our equations for F and G, we can already tell that there are values for A and B. So in order for us to plot these first, we can plot F by using what we know about Trigg gas, we can plot G and then we can add the Y coordinates to get our value for H of X. So let's start with F FX. So we know that F of X equals four multiplied by the cosine of two X. That already tells us that the amplitude for F FX is four and the period for F FX which is two pi divided by B becomes two pi divided by two, which would just be pi. So for F FX, we're going to draw the cosine graph, make it four times larger and shorten it to a period of pi. So let's go ahead and do that. So starting at, at four. OK. So we're gonna start at four and our period is a pie. So we're gonna go down to 1/4 of pie all the way down to half of pie. And then we're going to come back up. And that would be our first period. Let's do our next period by doing the same thing. OK. And that means our function for F FX would look something like that. Next. We need to draw our graph of G FX. What do we know about G FX? Well, G of X equals three multiplied by the cosine of X. And already that tells us that our amplitude is three and our period would be two pi because we've done nothing to the, the, the coefficient of X, the coefficient of X is one that tells us it's a regular period. So now we're going to draw another cosine graph, but this time it is going to be three times as large. So let's go ahead and see if we can do that. So keeping the same period of two pi start at three and we go down to pi negative three and then we're gonna come back up and then we're going to stop at three. Oops, I stopped at four. So let me just come back and try it properly. OK. So we come down and then we go up and then we stop at pike. So it's going to look something like that, know that we have our graph for F FX and G FX, as we said to get H FX, all you need to do is to add F and G. So we're going to do that for each of our points for each. Now when X equals zero, F FX is four, G FX is three. So that means H FX equals seven. Likewise, when, when, for the rest, I mean, you get the idea, you get the gist. So when X is 1/4 of pi, then our Y value would be 2.1 just a little bit above two there. OK? When X equals half of pi Y equals negative 4, 3/4 of pi Y equals negative 2.1. And when X equals py equals one, when X equals five fourths of pi Y equals negative 2.1. Again, when X equals three halves of pi Y equals 3/ of sorry, Y equals negative four. When X equals seven fourth, soft pi Y equals 2.1. And finally, when Y equals sorry, when X equals two pi Y equals seven. And again, you can check these in your calculator. As a matter of fact, you could load the function, you could just put hr X in your calculator as well and you're going to end up with those. So now that we have the points for age, let's plot. So I'm gonna go ahead and try to give it a nice, smooth, as smooth as it can be. And my plot would look something like that. So this would be the plot of H FX and those would be all of our functions on the same coordinate system. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)

Key Concepts

Graphing Trigonometric Functions

Function Addition and Pointwise Operations

Properties of Cosine Functions and Frequency

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