GRAPHS OF

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

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

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

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

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

Thinking about transformations that you learned 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

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

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)

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 )

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

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

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

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

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

2

2

32

542

2

3

…-3, -, , 3, . . .

Where is the function minimum?

0 22

Where does cos x = -1?

33

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)

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

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

The last thing we want to see is what happens if we put a coefficient on the x.

y = sin 2x

y = sin 2x

y = sin x

It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

What do you think will happen to the graph if we put a fraction in front?

y = sin 1/2 x

y = sin x

The period for one complete cycle is twice as long or 4

xy2

1sin

So if we look at y = sin x the affects the period.

The period T = 2 This will

be true for cosine as well.

What is the period of y = cos 4x?24

2 T

This means the graph will "cycle" every /2 or 4 times as often y = cos 4x

y = cos x

tAy cos tAy sin

absolute value of this is the amplitude

Period is 2 divided by this

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au