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

2 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 Functions 4

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Graphs of the Sine and Cosine Functions 4.1 Periodic Functions ▪ Graph of the Sine Function ▪ Graph of the Cosine Function ▪ Graphing Techniques, Amplitude, and Period ▪ Using a Trigonometric Model

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = sin bx (cont.) The x-values are

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = sin bx (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = cos bx (page 139) 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 x-values are 0, π, 2 π, 3 π, and 4 π.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = cos bx (page 139)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 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. Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 4 Graphing y = a sin bx (page 150)

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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 coefficient of x is, so b =, and the period is The x-values are 0, 1, 2, 3, and 4.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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 . The amplitude is The equation is

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Translations of the Graphs of the Sine and Cosine Functions 4.2 Horizontal Translations ▪ Vertical Translations ▪ Combinations of Translations ▪ Determining a Trigonometric Model Using Curve Fitting

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = sin ( x – d ) (cont.) Step 3: Evaluate the function for each of the five x-values.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = sin ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = sin ( x – d ) (cont.) 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a cos ( x – d ) (page 150) Step 2: Divide the period into four equal intervals: Step 1: b = 1, so find the interval whose length is

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a cos ( x – d ) (cont.) Step 3: Evaluate the function for each of the five x-values.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a cos ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cos b ( x – d ) (page 150) Write the equation in the form y = a cos b(x – d). Step 1: b = 2, so find the interval whose length is

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cos b ( x – d ) (cont.) Step 2: Divide the period into four equal intervals: Step 3: Evaluate the function for each of the five x-values.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cos b ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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. Step 2: Divide the period into four equal intervals:

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 4 Graphing y = c + a cos bx (cont.) Step 3: Evaluate the function for each of the five x-values.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 4 Graphing y = c + a cos bx (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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 =

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Graphs of the Tangent and Cotangent Functions 4.3 Graph of the Tangent Function ▪ Graph of the Cotangent Function ▪ Graphing Techniques

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = tan bx (page 162) Step 1: Find the period and locate the vertical asymptotes. Period = The asymptotes have the form

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = tan bx (cont.) Step 2: Sketch the two vertical asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = tan bx (cont.) 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.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = tan bx (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = tan bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a tan bx (page 163) Step 1: Find the period and locate the vertical asymptotes. The asymptotes have the form

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a tan bx (cont.) Step 2: Sketch the two vertical asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a tan bx (cont.) 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.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a tan bx (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a tan bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cot bx (page 163) Step 1: Find the period and locate the vertical asymptotes. The asymptotes have the form bx = 0 and bx = π.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cot bx (cont.) Step 2: Sketch the two vertical asymptotes.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cot bx (cont.) 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.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cot bx (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Graphing y = a cot bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

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

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 Graphs of the Secant and Cosecant Functions 4.4 Graph of the Secant Function ▪ Graph of the Cosecant Function ▪ Graphing Techniques ▪ Addition of Ordinates ▪ Connecting Graphs with Equations

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = a sec bx (page 172) Step 1: Graph the corresponding reciprocal function y = 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:

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = a sec bx (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Graphing y = a sec bx (cont.) Step 3: Sketch the graph.

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a csc ( x – d ) (cont.) Dividing the interval into four equal parts gives the key points:

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a csc ( x – d ) (cont.) Step 2: The vertical asymptotes of are at the x-intercepts of Continuing this pattern to the left, there are also vertical asymptotes at

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Graphing y = a csc ( x – d ) (cont.) Step 3: Sketch the graph.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3a Determining an Equation for a Graph (page 173) Determine an equation for the graph. Graph of y = sec x with period of 4 .

Copyright © 2013, 2009, 2005 Pearson Education, Inc 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