In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x)=sin(2 x)+

In Problems 25-32, use the method of adding y-coordinates to...

Key Concepts

Periodicity and Frequency

Each trigonometric function has a specific period, which is the interval over which the function repeats its values. When functions with different frequencies are added together, the interaction of their periods determines the overall periodic behavior of the resulting function.

Graphing Trigonometric Functions

Trigonometric functions, such as sine and cosine, have distinctive characteristics like amplitude, period, phase shift, and vertical shift. Understanding these features is essential, particularly when they appear in composite functions, as their periodic nature and oscillatory behavior influence the final graph.

Y-Coordinate Addition Method

This method involves computing the y-values of each individual function at the same x-coordinates and then summing these values to obtain the y-value of the composite function. It provides a systematic approach for graphing functions that are expressed as the sum of two or more component functions.

Graphing Composite Functions

Composite functions, especially those represented as the sum of two ordinary functions, can be analyzed by graphing each component separately and then combining their y-values. This technique is fundamental in understanding how individual function behaviors contribute to the overall shape of a more complex function.

Transcript

00:01 Okay, so we want to sketch this graph right here, sine 2x plus cosine x.

00:08 So the first thing we want to do is we want to see what each individual part looks like first.

00:12 So we want to see what the sign part looks like, and we want to see what the cosine part looks like on top of each other.

00:21 And that'll give us an indication of what the final graph of sine 2x plus cosine x looks like.

00:28 So let's start with the sign.

00:30 Part.

00:31 So the only thing we see that's affected here is the period.

00:34 So we need to figure out what the period of sine 2x is.

00:38 And if you remember to do that, you take, let's do green.

00:42 You take 2 pi and divide it by whatever is being multiplied on the inside.

00:50 And that's a 2.

00:51 So it's 2 pi over 2.

00:53 And so that means our period is pi.

00:56 So that means for our sign part, we're going to see one whole period of sign at pi.

01:02 And then we'll see another one from pi to two pi.

01:06 So let's sketch that.

01:08 So we'll start at zero, go up here, not go up here, and then back down to zero at pi over two, and then negative one right here, and then back to zero at pi.

01:24 Okay, and then we'll do that one more time.

01:34 Okay, and so now let's kind of fill it in with some dots to see what that looks like.

01:56 Okay, so that's what the...

01:58 The sign portion of this function looks like by itself.

02:03 Now let's sketch the cosine curve.

02:07 Okay, well this is just cosine without anything being affected to it.

02:10 So it's just the base curve for cosine.

02:12 And we know that starts at 1, it's 1 at 0, it's 0 at pi over 2, it's negative 1 at pi, 0 at 3 pi over 2, and then 1 at 2 pi.

02:28 Now let's kind of fill in the missing pieces with dots.

02:41 This is just the base curve of cosine, this blue part.

02:50 Okay, and that's what these two graphs look like on top of each other.

02:56 So just looking at the graph, there's certain places where we know where the whole function, which we'll do in red.

03:05 We know there's, we can go ahead and pick out certain spots of the whole function.

03:11 One of those spots is right here at the, at zero, because we know our sign of 2x is zero, and our cosine of x is 1.

03:23 And 0 plus 1 is 1.

03:25 So at 0, the overall function is 1.

03:29 And we can put a dot right there.

03:31 I'll put a big dot.

03:33 So we know that's for the whole function.

03:36 Okay, another place like this would be right here at pi over two.

03:41 Both the sign portion and the cosine portion are zero.

03:45 So we know that zero plus zero is zero.

03:48 So we see that we have a zero there.

03:52 We don't really know what it's doing right here.

03:55 We can probably guess that since we're both sides of sine and cosine are positive here, it might be going up a little further, but we don't know for sure, so we'll come back to that.

04:08 Let's keep looking for spots where we know for sure where it is.

04:12 If we keep going, we know that here, when the sign portion is zero, the cosine portion is negative 1.

04:23 And we know that 0 plus negative 1 is negative 1.

04:27 So we see that our whole function is negative 1 there.

04:32 Okay, if we keep going, we see that at 3 pi over 2, both of them are 0, so our whole function is 0 here.

04:39 And if we keep going, we see that at 2 pi, the sine function is 0, and the cosine function is 1, so our whole function is 1, right there.

04:53 Now, if we just look at these dots, it kind of looks like our function is completely following the cosine curve.

05:02 However, we know that's not the case because our sign of 2x is affecting it.

05:09 Like right here, we know that since these are both pretty big negatives, it's going to double the length down.

05:17 It's going to be probably somewhere down here.

05:19 But to do that, we'll check our points at pi over 4, 3 pi over 4, 5 pi over 4, and 7.

05:38 In pi over four.

05:40 These are just the in -between positions.

05:43 Okay, and i have a chart ready for us to do that.

05:46 Oh, well, we need to test sine of 2x and cosine of x.

05:58 Okay, so let's plug in pi over four.

06:02 When you plug in pi over four into this, you get sign of pi over two.

06:12 And we know that sine of pi over two is one.

06:20 Okay, now let's plug in to cosine of x.

06:24 We know that cosine of pi over four, if you remember what pi over four is in degrees, that's 45 degrees...

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