copyright © 2009 pearson addison-wesley 4.2-1 4 graphs of the circular functions

Copyright © 2009 Pearson Addison-Wesley 4.2-1

4Graphs 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

4.5 Harmonic Motion

4Graphs of the Circular Functions

Copyright © 2009 Pearson Addison-Wesley

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4.2-3

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


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.


Horizontal Translations

In the function y = f(x – d), the expression x – d is called the argument.

A horizontal translation is called a phase shift.


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Example 1 GRAPHING y = sin (x – d)

Graph over one period.

Divide this interval into four equal parts:

To find an interval of one period, solve the three-part

inequality

Method 1


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Example 1 GRAPHING y = sin (x – d) (continued)

Make a table of values determined by the x-values.


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Join the corresponding points with a smooth curve.

The period is 2π, and the amplitude is 1.


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The argument indicates that the graph of

y = sin x will be translated units to the right.

Method 2


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Example 2 GRAPHING y = a cos (x – d)


Method 1


inequality


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Example 2 GRAPHING y = a cos (x – d) (continued)



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Join the corresponding points with a smooth curve.

The period is 2π, and the amplitude is 3.


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Method 2

so the phase

shift is units to the left.


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Example 3 GRAPHING y = a cos b(x – d)


Method 1


inequality


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Example 3 GRAPHING y = a cos b(x – d) (continued)



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Join the corresponding points with a smooth curve. Then graph an additional half-period to the left and to the right.


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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.


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.


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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


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Example 4 GRAPHING y = c + a cos bx (continued)

Make a table of points.


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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.


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Guidelines for Sketching Graphs of Sine and Cosine Functions

Step 2 Divide the interval into four equal parts.

Method 1

Step 1 Find an interval whose length is one period by solving the three-part inequality

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 pointsthat intersect the line y = c.


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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.


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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.


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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.


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Example 5 GRAPHING y = c + a sin b(x – d) (cont.)

Step 3: Make a table of values.

Step 2: Divide the interval into four equal parts to get the x-values.


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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.


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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.


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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.


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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,


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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.


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(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.


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(c) Use the sine regression featrue of a graphing calculator to determine a second model for these data.

copyright © 2009 pearson addison-wesley 4.2-1 4 graphs of the circular functions

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