If you have not watched the PowerPoint on the unit circle you should watch it first. After youve...

28
If you have not watched the PowerPoint on the unit circle you should watch it first. After you’ve watched that PowerPoint you are ready for this one. If you watched it, just click to begin this part of the section.

Transcript of If you have not watched the PowerPoint on the unit circle you should watch it first. After youve...

Page 1: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

If you have not watched the PowerPoint on the unit circle you should watch it first. After you’ve watched that PowerPoint you are ready for this one.

If you watched it, just click to begin this part of the section.

Page 2: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Let’s think about the function y = sin x

What is the domain? (remember domain means the “legal” things you can put in for x ). You can put in anything you want

so the domain is all real numbers.

What is the range? (remember range means what you get out of the function). The range is: -1 sin x 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?)

Page 3: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Let’s think about the function y = cos x

What is the domain? (remember domain means the “legal” things you can put in for x). You can put in anything you want

so the domain is all real numbers.

What is the range? (remember range means what you get out of the function). The range is: -1 cos x 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?)

Page 4: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

2

3,

2

1

Look at the unit circle and determine sin 420°.

All the way around is 360° so we’ll need more than that. We see that it will be the same as sin 60° since they are coterminal angles. So sin 420° = sin 60°.

In fact sin 780° = sin 60° since that is just another 360° beyond 420°.

Because the sine values are equal for coterminal angles that are multiples of 360° added to an angle, we say that the sine is periodic with a period of 360° or 2.

Page 5: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

2

3,

2

1

The cosine is also periodic with a period of 360° or 2.

1

Page 6: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

GRAPHS OF

Page 7: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

We are interested in the graph of y = f(x) = sin x

Start with a "t" chart and let's choose values from our unit circle and find the sine values.

x y = sin x

6

0 0

2

1

2

1

6

52

1We are dealing with x's and y's on the unit circle to find values. These are completely different from the x's and y's used here for our function.

x

y

1

- 1

plot these points

Page 8: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

y = f(x) = sin xchoose more values

x y = sin x

6

7 0

2

1

2

31

6

112

1

If we continue picking values for x we will start to repeat since this is periodic.

x

y

1

- 1

plot these points

2 0

join the points

6

2

Page 9: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2.

It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function)

2 22 2

Page 10: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

From College Algebra recall that an odd function (which the sine is) is symmetric with respect to the origin as can be seen here

Page 11: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

What are the x intercepts? Where does sin x = 0?

0 2 3 423

…-3, -2, -, 0, , 2, 3, 4, . . .

Where is the function maximum? Where does sin x = 1?

2

2

52

3

2

7

2

5,

2,

2

3,

2

7

Page 12: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Where is the function minimum? Where does sin x = -1?

0 2 3 423

2

2

52

3

2

7

2

7,

2

3,

2,

2

5

2

5

2

2

32

7

Page 13: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Thinking about transformations that you learned in College Algebra and knowing what y = sin x looks like, what do you suppose y = sin x + 2 looks like?

The function value (or y value) is just moved up 2.

y = sin x

y = 2 + sin x This is often written with terms traded places so as not to confuse the 2 with part of sine function

Page 14: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Thinking about transformations that you've learned and knowing what y = sin x looks like, what do you suppose y = sin x - 1 looks like?

The function value (or y value) is just moved down 1.

y = sin x

y = - 1 + sin x

Page 15: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin (x + /2) looks like?

This is a horizontal shift by - /2

y = sin x

y = sin (x + /2)

Page 16: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = - sin (x )+1 looks like?

This is a reflection about the x axis (shown in green) and then a vertical shift up one.

y = sin x

y = - sin x

y = 1 - sin (x )

Page 17: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

What would the graph of y = f(x) = cos x look like?We could do a "t" chart and let's choose values from our unit circle and find the cosine values.

x y = cos x

3

0 1

2

1

2

0

3

22

1 We could have used the same values as we did

for sine but picked ones that gave us easy values to plot.

x

y

1

- 1

plot these points

6

Page 18: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

y = f(x) = cos x Choose more values.

x y = cos x

3

4 1

2

1

2

30

3

52

1

cosine will then repeat as you go another loop around the unit circle

x

y

1

- 1

plot these points

6

2 1

Page 19: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Here is the graph y = f(x) = cos x showing from -2 to 6. Notice it repeats with a period of 2.

It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function)

2 22 2

Page 20: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

Page 21: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

What are the x intercepts? Where does cos x = 0?

2

2

32

52

2

3

…-4, -2, , 0, 2, 4, . . .

Where is the function maximum? Where does cos x = 1?

0 22

2

5,

2

3,

2,

2,

2

3

Page 22: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

2

2

32

542

2

3

…-3, -, , 3, . . .

Where is the function minimum?

0 22

Where does cos x = -1?

33

Page 23: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

You could graph transformations of the cosine function the same way you've learned for other functions.

Let's try y = 3 - cos (x - /4)

reflects over x axis

moves up 3 moves right /4

y = cos x y = - cos x

y = 3 - cos x y = 3 - cos (x - /4)

Page 24: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

These graphs illustrate how as you go around the unit circle and plot the y value (upper left) or the x value (lower right) you generate the sine and cosine graphs. The lower left shows the value of the angle t at any given time. Notice the axis for cosine are reversed here so you can see how the x value moves but you can rotate this graph to have t horizontal and x vertical and see the cosine graph like it is traditionally graphed.

y = sin t

unit circle

x = cos tvalue of angle t

Page 25: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

What would happen if we multiply the function by a constant?

y = 2 sin x

All function values would be twice as high

y = 2 sin x

y = sin x

The highest the graph goes (without a vertical shift) is called the amplitude.

amplitude of this

graph is 2

amplitude is here

Page 26: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

For y = A cos x and y = A sin x, A is the amplitude.

y = 4 cos x y = -3 sin x

What is the amplitude for the following?

amplitude is 4 amplitude is 3

Page 27: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

cosy A x C D siny A x C D

absolute value of this is the amplitude

This is the phase shift (horizontal translation)

remember it is opposite in sign

This is the vertical translation

Page 28: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If.

Given this graph, let’s see if we can find it’s equation in the form y = A sin (x – C)+ D

First let’s find the vertical center of the graph and then we can determine the amplitude.

ASo what is A? A = 2

Now let’s determine the vertical shift D.

D

So what is D?

D = 3

There is no horizontal shift so C = 0 and we have our equation.

y = 2 sin (x) + 3