Problem are based on material tearned earlier in the course....

Problem are based on material tearned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the amplitude, period, and phase shift of $y=\frac{3}{2} \cos (6 x+3 \pi) .$ Graph the function, showing at least two periods.

Problem are based on material tearned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Find the amplitude, period, and phase shift of $y=\frac{3}{2} \cos (6 x+3 \pi) .$ Graph the function, showing at least two periods.

Key Concepts

Graphical Transformation

Graphing a trigonometric function, such as a cosine function, involves applying vertical stretches (amplitude), horizontal stretches (period), and horizontal translations (phase shift) to the basic cosine curve. This transformation process is essential to accurately represent the function’s behavior over its cycle and to ensure the graph reflects the correct features over multiple periods.

Amplitude

The amplitude of a trigonometric function indicates its vertical stretch or compression. It is determined by the absolute value of the coefficient in front of the cosine or sine term. In the context of the cosine function, the amplitude represents the maximum displacement from the midline, and is a measure of the wave’s height.

Period

The period of a trigonometric function is the length (in the x-direction) of one complete cycle of the wave. It can be calculated by taking 2? divided by the absolute value of the coefficient of x inside the cosine or sine function. This determines how frequently the wave oscillates over its domain.

Phase Shift

The phase shift of a trigonometric function refers to the horizontal translation of the graph, indicating where the standard cycle of the function begins. For a function in the form cosine or sine, the phase shift is determined by the horizontal component added or subtracted inside the function's argument. It is calculated by rewriting the argument in a form that factors out the coefficient of x, and then isolating the horizontal translation.

Transcript

00:01 We have a trig function, y equals three halves, cosine, of 6x plus three pi.

00:08 And i don't know if your textbook has you do this, but i'm going to factor out that six.

00:16 And when you factor, it's like you divide each of these by six, and when i divide that, i end up getting pi over two.

00:25 So now i can see that my amplitude is three halves or one and a half.

00:32 I can see that my period is equal to 2 pi divided by this coefficient, divided by 6, so it's pi over 3.

00:42 I have a phase shift of negative pi over 2, and so i'm going to scale my graph.

00:52 This graph is going to be very short.

00:53 We need to graph only pi over 3.

00:57 So if you saw the graph between 0 and 2 pi, you would see six cycles.

01:04 Six cycles between zero and pi to pi.

01:11 Now, you aren't asked to use the zero to two pi.

01:16 You just have to graph two cycles.

01:18 So let's go through and we're going to scale this graph like so.

01:22 And another thing i like to do, i don't know that your class has you do this, but i like to find what i call the critical points.

01:29 And that happens every one -fourth of the period.

01:32 Every one -fourth of the period, you're going to go every pi over 12, you're going to start at a high, and then you're going to move to a zero, and then you're going to move to a low, and then to a zero, and to a high.

01:44 So our graph has to go up to, here's one, here's two, here's one and a half.

01:51 We'll go down to, here's negative one, negative two, and we need to go down to negative one and a half.

01:59 So our graph, click scale that, where our graph is going to stay between positive one and a half, and negative 1 .5 and we have to graph at least two cycles.

02:14 And so i'm going to go back.

02:15 Here's 1, 2, 3, 4, 5, 6.

02:19 There is negative pi over 2.

02:23 And then 1, 2, 3, 4, 5, 6...

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