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

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) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Accepted Answer

Welcome back everyone. In this problem, we want to plot the functions FG and H in the same coordinate system for X being between zero and two pi by adding or subtracting the Y coordinates for our functions F FX equals two multiplied by the sin of two XG FX equals three, multiplied by the cosine of X and H of X equals F minus G of X. Now, what do we know? Well, we want to plot it in the same coordinate system. We already know what F FX and G FX is equal to and no, to find H of X, we need to use this function. What recall that F minus G FX is the same thing as F FX minus G FX, right. So that means H of X would be equal to F FX which is two multiplied by the sin of two X minus G FX, which is three multiplied by the cosine of X. So to find H FX, what it's really saying is that we're going to find the difference between F FX and G FX. So that means to get it, we're going to subtract the Y coordinates because F FX and G FX represent the Y coordinate. So since we already have a graph for F FX, a S sign graph and a cosine graph for GEO, if we go ahead and plot those, we can figure out what H FX is going to do look like because we can also recall that for every trick function it's written in the form a multiplied by that trick function, whether it's sine or cosine of BX minus C plus D. And if you look at both of our equations, we can tell what our values of A and B will be. So let's start with F FX. And remember while we're working A is our amplitude. OK. And or we use B to help us find a period of the graph because a period is two pi divided by B. So let's start with F FX. So given F of X equals two multiplied by the sine of two XX. That tells us that our amplitude equals the coefficient of the sine uh the, the, the sine graph which is two and our period equals two pi divided by B. In this case, B equals two. So that's two pi divided by two, which would just be equal to pi. So for our function F, we're going to draw a sand graph that's twice as large with a period of pi. So let's go ahead and do that for our sand graph. Remember we start at 00, come back to our region, we go down to the minimum of negative two and come back to pi that would be for the first period, for the second period. Let's go again. So we're drawing that and, or, or, or a sign graph would look something like that. OK. Next, we want to draw our graph for three for geographics. So given G FX equals three, multiplied by the cosine of X. Well, here this graph right away, we realize that our amplitude is three. OK. And our period would be equal to two pi because there's no coefficient of X there. So it'll be two pi divided by one which is two pi. So we're just drawing our regular cosine graph but making it three times larger. So let's go ahead and drop that one. So starting at three, we're going to come down or we're gonna go down to negative three for pi, then we're gonna come back up and then we're going to stop at two pi. So that is what our graph of G FX would look like. Know that we have the graphs for G FX and F FX respectively. We can go ahead and plot our graph for H FX because remember it's the difference between the Y and X coordinates. It's F FX minus G FX when X equals zero H FX would be equal to F FX, which is zero minus G FX which is three. And that gives us negative three when X equals 1/4 of pi. It would be 2.1 minus two which gives us negative 0.1 which would be a little bit below our X axis. When X, when X is half of pi Y would be equal to zero when X is 3/4 soft pi or would be equal to 0. when it's equal to pi Y would be equal to three. When it's equal to a five fourth soft pi Y would be equal to 4.1. When it's, when it's equal to three hives of pi Y would be equal to zero, it cancels out there when it's seven fourths of pi Y equals negative 4.1. And you can double check these with your own uh calculator by putting this function into your calculator. So it would be negative 4.1 and when X is two, pi Y is negative three. So now we have the points for our graph, we can go ahead and plot and it's going to look something like this still a curve because it's a combination of sine and cosine graphs. And that would be our curve for our graph for H FX. 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) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Key Concepts

Trigonometric Functions

Graphing Functions

Function Addition and Subtraction

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