4 Graphs of the Circular Functions Copyright © 2009 Pearson Addison-Wesley

Graphs of the Circular Functions 4 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 4.5 Harmonic Motion Copyright © 2009 Pearson Addison-Wesley

Translations of the Graphs of the Sine and Cosine Functions 4.2 Translations of the Graphs of the Sine and Cosine Functions Horizontal Translations ▪ Vertical Translations ▪ Combinations of Translations ▪ Determining a Trigonometric Model Using Curve Fitting Copyright © 2009 Pearson Addison-Wesley 1.1-3

Horizontal Translations The graph of the function defined by y = f(x – d) is translated horizontally compared to the graph of y = f(x). The translation is d units to the right if d > 0 and |d| units to the left if d < 0. Copyright © 2009 Pearson Addison-Wesley

Horizontal Translations A horizontal translation is called a phase shift. In the function y = f(x – d), the expression x – d is called the argument. Copyright © 2009 Pearson Addison-Wesley

To find an interval of one period, solve the three-part inequality Example 1 GRAPHING y = sin (x – d) Graph over one period. To find an interval of one period, solve the three-part inequality Method 1 Divide this interval into four equal parts: Copyright © 2009 Pearson Addison-Wesley 1.1-6

Make a table of values determined by the x-values. Example 1 GRAPHING y = sin (x – d) (continued) Make a table of values determined by the x-values. Copyright © 2009 Pearson Addison-Wesley 1.1-7

The period is 2π, and the amplitude is 1. Example 1 GRAPHING y = sin (x – d) (continued) Join the corresponding points with a smooth curve. The period is 2π, and the amplitude is 1. Copyright © 2009 Pearson Addison-Wesley 1.1-8

Example 1 GRAPHING y = sin (x – d) (continued) Method 2 The argument indicates that the graph of y = sin x will be translated units to the right. Copyright © 2009 Pearson Addison-Wesley 1.1-9

To find an interval of one period, solve the three-part inequality Example 2 GRAPHING y = a cos (x – d) Method 1 To find an interval of one period, solve the three-part inequality Divide this interval into four equal parts: Copyright © 2009 Pearson Addison-Wesley 1.1-10

Make a table of values determined by the x-values. Example 2 GRAPHING y = a cos (x – d) (continued) Make a table of values determined by the x-values. Copyright © 2009 Pearson Addison-Wesley 1.1-11

The period is 2π, and the amplitude is 3. Example 2 GRAPHING y = a cos (x – d) (continued) Join the corresponding points with a smooth curve. The period is 2π, and the amplitude is 3. Copyright © 2009 Pearson Addison-Wesley 1.1-12

so the phase shift is units to the left. Example 2 GRAPHING y = a cos (x – d) (continued) Method 2 so the phase shift is units to the left. Copyright © 2009 Pearson Addison-Wesley 1.1-13

To find an interval of one period, solve the three-part inequality Example 3 GRAPHING y = a cos b(x – d) Method 1 To find an interval of one period, solve the three-part inequality Divide this interval into four equal parts: Copyright © 2009 Pearson Addison-Wesley 1.1-14

Make a table of values determined by the x-values. Example 3 GRAPHING y = a cos b(x – d) (continued) Make a table of values determined by the x-values. Copyright © 2009 Pearson Addison-Wesley 1.1-15

Example 3 GRAPHING y = a cos b(x – d) (continued) Join the corresponding points with a smooth curve. Then graph an additional half-period to the left and to the right. Copyright © 2009 Pearson Addison-Wesley 1.1-16

Write the expression in the form a cos b(x – d). Example 3 GRAPHING y = a cos b(x – d) (continued) Method 2 Write the expression in the form a cos b(x – d). Then a = –2, b = 3, and The amplitude is |–2| = 2, the period is and the phase shift is units to the left as compared to the graph of y = –2 cos 3x. Copyright © 2009 Pearson Addison-Wesley 1.1-17

Vertical Translations The graph of the function defined by y = c + f(x) is translated vertically compared to the graph of y = f(x). The translation is c units up if c > 0 and |c| units down if c < 0. Copyright © 2009 Pearson Addison-Wesley

The period of –2 cos 3x is so the key points have x-values Example 4 GRAPHING y = c + a cos bx The graph of y = 3 – 2 cos 3x is the same as the graph of y = –2 cos 3x, vertically translated 3 units up. The period of –2 cos 3x is so the key points have x-values Copyright © 2009 Pearson Addison-Wesley 1.1-19

Example 4 Make a table of points. GRAPHING y = c + a cos bx (continued) Make a table of points. Copyright © 2009 Pearson Addison-Wesley 1.1-20

Example 4 GRAPHING y = c + a cos bx (continued) Join the corresponding points with a smooth curve. Then graph an additional period to the left. Copyright © 2009 Pearson Addison-Wesley 1.1-21

Guidelines for Sketching Graphs of Sine and Cosine Functions Method 1 Step 1 Find an interval whose length is one period by solving the three-part inequality Step 2 Divide the interval into four equal parts. Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c. Copyright © 2009 Pearson Addison-Wesley 1.1-22

Guidelines for Sketching Graphs of Sine and Cosine Functions Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. Step 5 Draw the graph over additional periods as needed. Copyright © 2009 Pearson Addison-Wesley 1.1-23

Guidelines for Sketching Graphs of Sine and Cosine Functions Method 2 First graph y = a sin bx or y = a cos bx. The amplitude of the function is |a|, and the period is Use translations to graph the desired function. The vertical translation is c units up if c > 0 and |c| units down if c < 0. The horizontal translation (phase shift) is d units to the right if d > 0 and |d| units to the left if d < 0. Copyright © 2009 Pearson Addison-Wesley 1.1-24

Step 1: Find an interval whose length is one period. Example 5 GRAPHING y = c + a sin b(x – d) Use Method 1: Step 1: Find an interval whose length is one period. Divide each part by 4. Subtract from each part. Copyright © 2009 Pearson Addison-Wesley 1.1-25

Step 2: Divide the interval into four equal parts to get the x-values. Example 5 GRAPHING y = c + a sin b(x – d) (cont.) Step 2: Divide the interval into four equal parts to get the x-values. Step 3: Make a table of values. Copyright © 2009 Pearson Addison-Wesley 1.1-26

Example 5 GRAPHING y = c + a sin b(x – d) (cont.) Copyright © 2009 Pearson Addison-Wesley 1.1-27

Example 5 GRAPHING y = c + a sin b(x – d) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve. Extend the graph an additional half-period to the left and to the right to include two full periods. Copyright © 2009 Pearson Addison-Wesley 1.1-28

Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION The maximum average monthly temperature in New Orleans is 82°F and the minimum is 54°F. The table shows the average monthly temperatures. Copyright © 2009 Pearson Addison-Wesley 1.1-29

Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION (continued) The scatter diagram for a two-year interval suggests that the temperatures can be modeled with a sine curve. (a) Using only the maximum and minimum temperatures, determine a function of the form where a, b, c, and d are constants, that models the average monthly temperature in New Orleans. Let x represent the month, with January corresponding to x = 1. Copyright © 2009 Pearson Addison-Wesley 1.1-30

The average of the maximum and minimum temperatures gives c: Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION (continued) Use the maximum and minimum average monthly temperatures to determine the amplitude a: The average of the maximum and minimum temperatures gives c: Since temperatures repeat every 12 months, Copyright © 2009 Pearson Addison-Wesley 1.1-31

Trial and error using a calculator leads to d = 4.2. Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION (continued) To determine the phase shift, observe that the coldest month is January, when x = 1, and the hottest month is July, when x = 7. So choose d to be about 4. The table shows that temperatures are actually a little warmer after July than before, so experiment with values just greater than 4 to find d. Trial and error using a calculator leads to d = 4.2. Copyright © 2009 Pearson Addison-Wesley 1.1-32

The graph shows the data points from the table, the graph of Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION (continued) (b) On the same coordinate axes, graph f for a two-year period together with the actual data values found in the table. The graph shows the data points from the table, the graph of and the graph of for comparison. Copyright © 2009 Pearson Addison-Wesley 1.1-33

Example 6 MODELING TEMPERATURE WITH A SINE FUNCTION (continued) (c) Use the sine regression featrue of a graphing calculator to determine a second model for these data. Copyright © 2009 Pearson Addison-Wesley 1.1-34