Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Graphs of the Circular Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

4

Graphs of the Circular Functions



4.1 Graphs of the Sine and Cosine Functions

4.2 Translations of the Graphs of the Sine and Cosine Functions

4.3 Graphs of the Tangent and Cotangent Functions

4.4 Graphs of the Secant and Cosecant Functions

Graphs of the Circular Functions4


Graphs of the Sine and Cosine Functions4.1Periodic Functions ▪ Graph of the Sine Function▪ Graph of the Cosine Function ▪ Graphing Techniques, Amplitude, and Period ▪ Using a Trigonometric Model


and compare to the graph of y = sin x.

4.1 Example 1 Graphing y = a sin x (page 137)

The shape of the graph is the same as the shape of y = sin x.

The range of is


and compare to the graph of y = sin x.

4.1 Example 2 Graphing y = sin bx (page 138)

The coefficient of x is , so b = , and the period is

Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts.

The endpoints are 0 and , and the three points between the endpoints are


4.1 Example 2 Graphing y = sin bx (cont.)

The x-values are


4.1 Example 2 Graphing y = sin bx (cont.)


4.1 Example 3 Graphing y = cos bx (page 139)



The x-values are 0, π, 2π, 3π, and 4π.


4.1 Example 3 Graphing y = cos bx (page 139)


Graph y = –3 sin 2x.

4.1 Example 4 Graphing y = a sin bx (page 140)

The coefficient of x is 2, so b = 2, and the period is

The amplitude is |–3| = 3.



4.1 Example 4 Graphing y = a sin bx (page 150)


4.1 Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (page 141)

The amplitude is |2| = 2.

Divide the interval [0, 4] into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts.


The x-values are 0, 1, 2, 3, and 4.


4.1 Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (cont.)


4.1 Example 6 Determining an Equation for a Graph (page 141)

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0 for the given graph.

The graph is that of a sine function with period .

2 22b

b

The amplitude is

1 1 1 1 1(1)

2 2 2 2 2

The equation is 1

sin2 .2

y x


Translations of the Graphs of the Sine and Cosine Functions4.2Horizontal Translations ▪ Vertical Translations ▪ Combinations of Translations ▪ Determining a Trigonometric Model Using Curve Fitting


4.2 Example 1 Graphing y = sin (x – d) (page 149)

Step 2: Divide the period into four equal intervals:

Step 1: b = 1, so find the interval whose length is


4.2 Example 1 Graphing y = sin (x – d) (cont.)

Step 3: Evaluate the function for each of the five x-values.



Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.



Note that this is the graph of y = sin x translated units to the left.

Amplitude = 1

Period =

Phase shift = units left

No vertical translation


4.2 Example 2 Graphing y = a cos (x – d) (page 150)




4.2 Example 2 Graphing y = a cos (x – d) (cont.)







Note that this is the graph of y = –2 cos x translated units to the right.

Amplitude = 2

Period = No vertical translation

Phase shift = units right


4.2 Example 3 Graphing y = a cos b(x – d) (page 150)

Write the equation in the form y = a cos b(x – d).



4.2 Example 3 Graphing y = a cos b(x – d) (cont.)








Amplitude, b = 2, so the period is

Phase shift, units to the right as compared

to the graph of


4.2 Example 4 Graphing y = c + a cos bx (page 151)

Step 1: Find the interval whose length is one period,

Graph y = –2 + 3 cos 2x over two periods.



4.2 Example 4 Graphing y = c + a cos bx (cont.)







Note that this is the graph of y = 3 cos 2x translated 2 units down.

Amplitude = 2

Phase shift : none

Vertical translation: 2 units down

Period =


Graphs of the Tangent and Cotangent Functions4.3Graph of the Tangent Function ▪ Graph of the Cotangent Function ▪ Graphing Techniques


4.3 Example 1 Graphing y = tan bx (page 162)

Step 1: Find the period and locate the vertical asymptotes.

Period =

The asymptotes have the form


4.3 Example 1 Graphing y = tan bx (cont.)

Step 2: Sketch the two vertical asymptotes.



Step 3: Divide the interval into four equal intervals.

Step 4: Evaluate the function for the three middle x-values to find the first-quarter point, midpoint, and third-quarter point.





Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.


4.3 Example 2 Graphing y = a tan bx (page 163)


The asymptotes have the form


4.3 Example 2 Graphing y = a tan bx (cont.)












4.3 Example 3 Graphing y = a cot bx (page 163)


The asymptotes have the form bx = 0 and bx = π.


4.3 Example 3 Graphing y = a cot bx (cont.)












4.3 Example 4 Graphing a Tangent Function With a Vertical Translation (page 164)

Every value of this graph will be 3 units less than the corresponding value of y in y = tan x, causing the graph of y = –3 + tan x to be translated 3 units down compared with the graph of y = tan x.


Graphs of the Secant and Cosecant Functions4.4Graph of the Secant Function ▪ Graph of the Cosecant Function ▪ Graphing Techniques ▪ Addition of Ordinates ▪ Connecting Graphs with Equations


4.4 Example 1 Graphing y = a sec bx (page 172)

Step 1: Graph the corresponding reciprocal functiony = 3 cos 2x.

Graph y = 3 sec 2x.

One period is in the interval Dividing the interval into four equal parts gives the key points:


4.4 Example 1 Graphing y = a sec bx (cont.)



Step 2: The vertical asymptotes of y = 3 sec 2x are at the x-intercepts of y = 3 cos 2x.

Continuing this pattern to the left, there are also vertical

asymptotes at



Step 3: Sketch the graph.


4.4 Example 2 Graphing y = a csc (x – d) (page 172)

Step 1: Graph the corresponding reciprocal function

The phase shift is units left, so one period is in the

interval


4.4 Example 2 Graphing y = a csc (x – d) (cont.)

Dividing the interval into four equal parts gives the key points:



Step 2: The vertical asymptotes of are at

the x-intercepts of

Continuing this pattern to the left, there are also vertical

asymptotes at



Step 3: Sketch the graph.


4.4 Example 3a Determining an Equation for a Graph (page 173)

Determine an equation for the graph.

1sec

2y x

Graph of y = sec x with period of 4.


4.4 Example 3b Determining an Equation for a Graph (page 173)

Determine an equation for the graph.

This is the graph of y = csc x translated one unit down.

y = –1 + csc x

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Graphs of the Circular Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1