Trig m4 Handouts

1

Rev.S08

MAC 1114

Module 4

Graphs of the CircularFunctions

2Rev.S08

Learning Objectives

Upon completing this module, you should be able to:

1. Recognize periodic functions.2. Determine the amplitude and period, when given the equation

of a periodic function.3. Find the phase shift and vertical shift, when given the

equation of a periodic function.4. Graph sine and cosine functions.5. Graph cosecant and secant functions.6. Graph tangent and cotangent functions.7. Interpret a trigonometric model.

http://faculty.valenciacc.edu/ashaw/Click link to download other modules.

3Rev.S08

Graphs of the Circular Functions


- Graphs of the Sine and Cosine Functions- Graphs of the Sine and Cosine Functions- Translations of the Graphs of the Sine and Cosine- Translations of the Graphs of the Sine and CosineFunctionsFunctions- Graphs of the Other Circular Functions- Graphs of the Other Circular Functions

There are three major topics in this module:

2

4Rev.S08

Introduction to Periodic Function


A periodic function is a function f such that f(x) = f(x + np),

for every real number x in the domain of f, everyinteger n, and some positive real number p. Thesmallest possible positive value of p is the periodof the function.

5Rev.S08

Example of a Periodic Function


6Rev.S08

Example of Another Periodic Function


3

7Rev.S08

What is the Amplitude of a PeriodicFunction?


The amplitude of a periodic function is half thedifference between the maximum and minimumvalues.

The graph of y = a sin x or y = a cos x, witha ≠ 0, will have the same shape as the graph ofy = sin x or y = cos x, respectively, except therange will be [−|a|, |a|]. The amplitude is |a|.

8Rev.S08

How to Graph y = 3 sin(x) ?


0−30303sin x

0−1010sin x

π3π/2ππ/20x

Note the difference between sin x and 3sin x. What is thedifference?

9Rev.S08

How to Graph y = sin(2x)?


The period is 2π/2 = π. The graph will completeone period over the interval [0, π].

The endpoints are 0 and π, the three middlepoints are:

Plot points and join in a smooth curve.

4

10Rev.S08

How to Graph y = sin(2x)?(Cont.)


Note the difference between sin x and sin 2x. What is thedifference?

11Rev.S08

Period of a Periodic Function


•Based on the previous example, we cangeneralize the following:For b > 0, the graph of y = sin bx willresemble that of y = sin x, but with period 2π/b.

The graph of y = cos bx will resemble that ofy = cos x, with period 2π/b.

12Rev.S08

How to Graph y = cos (2x/3)over one period?


The period is 3π. Divide the interval into four equal parts.

Obtain key points for one period.

10−101cos 2x/32π3π/2ππ/202x/33π9π/43π/23π/40x

5

13Rev.S08

How to Graph y = cos(2x/3)over one period? (Cont.)


The amplitude is 1. Join the points and connect with a smooth curve.

14Rev.S08

Guidelines for Sketching Graphs of Sineand Cosine Functions


To graph y = a sin bx or y = a cos bx, withb > 0, follow these steps.

Step 1 Find the period, 2π/b. Start with 0 onthe x-axis, and lay off a distance of2π/b.

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 x-intercepts.

15Rev.S08

Guidelines for Sketching Graphs of Sineand Cosine Functions Continued


Step 4 Plot the points found in Step 3, and jointhem with a sinusoidal curve havingamplitude |a|.

Step 5 Draw the graph over additional periods,to the right and to the left, as needed.

6

16Rev.S08

How to Graph y = −2 sin(4x)?


Step 1 Period = 2π/4 = π/2. The function willbe graphed over the interval [0, π/2] .

Step 2 Divide the interval into four equal parts.

Step 3 Make a table of values

020−20−2 sin 4x0−1010sin 4x

2π3π/2ππ/204x

π/23π/8π/4π/80x

17Rev.S08

How to Graph y = −2 sin(4x)? (Cont.)


Plot the points and join them with a sinusoidalcurve with amplitude 2.

18Rev.S08

What is a Phase Shift?


In trigonometric functions, a horizontal translationis called a phase shift.

In the equationthe graph is shifted π/2 units to the right.

7

19Rev.S08

How to Graph y = sin (x − π/3) by UsingHorizontal Translation or Phase Shift?


Find the interval for one period.

Divide the interval into four equal parts.

20Rev.S08

How to Graph y = sin (x − π/3) by UsingHorizontal Translation or Phase Shift?

(Cont.)


0−1010sin (x − π/3)2π3π/2ππ/20x − π/3

7π/311π/64π/35π/6π/3x

21Rev.S08

How to Graph y = 3 cos(x + π/4) by UsingHorizontal Translation or Phase Shift?


Find the interval.

Divide into four equal parts.

8

22Rev.S08

How to Graph y = 3 cos(x + π/4) by UsingHorizontal Translation or Phase Shift?


30−3033 cos (x + π/4)10−101cos(x + π/4)

2π3π/2ππ/20x + π/47π/45π/43π/4π/4−π/4x

23Rev.S08

How to Graph y = 2 − 2 sin 3x by UsingVertical Translation or Vertical Shift?


242022 − 2 sin 3x020−20−2 sin 3x

2π3π/2ππ/203x

2π/3π/2π/3π/60x

The graph is translated 2 units up from the graphy = −2 sin 3x.

24Rev.S08

How to Graph y = 2 − 2 sin 3x by UsingVertical Translation or Vertical Shift?

(Cont.)


Plot the points and connect.

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25Rev.S08

Further Guidelines for Sketching Graphsof Sine and Cosine Functions


Method 1: Follow these steps.Step 1 Find an interval whose length is one

period 2π/b by solving the three partinequality 0 ≤ b(x − d) ≤ 2π.

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. Thepoints will be maximum points,minimum points, and points that intersect theline y = c (middle points of the wave.)

26Rev.S08

Further Guidelines for Sketching Graphsof Sine and Cosine Functions (Cont.)


Step 4 Plot the points found in Step 3, and jointhem with a sinusoidal curve havingamplitude |a|.

Step 5 Draw the graph over additional periods,to the right and to the left, as needed.

Method 2: First graph the basic circular function. Theamplitude of the function is |a|, and the periodis 2π/b. Then use translations to graph thedesired function. The vertical translation is cunits up if c > 0 and |c| units down if c < 0.The horizontal translation (phase shift) is dunits to the right if d > 0 and |d| units to theleft if d < 0.

27Rev.S08

How to Graph y = −1 + 2 sin (4x + π)?


Write the expression in theform c + a sin b(x − d) byrewriting 4x + π as

Step 1

Step 2: Divide theinterval.

Step 3 Table

10

28Rev.S08

How to Graph y = −1 + 2 sin (4x + π)?(Cont.)


−1−3−11−1−1 + 2sin(4x + π)

2−20202 sin 4(x + π/4)

0−1010sin 4(x + π/4)

2π3π/2ππ/204(x + π/4)

π/23π/8π/4π/80x + π/4

π/4π/80−π/8−π/4x

29Rev.S08

How to Graph y = −1 + 2 sin (4x + π)?(Cont.)


Steps 4 and 5 Plot the points found in the table and join then with a

sinusoidal curve.

30Rev.S08

Let’s Take a Look at Other CircularFunctions.


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31Rev.S08

Cosecant Function


32Rev.S08

Secant Function


33Rev.S08

Guidelines for Sketching Graphs ofCosecant and Secant Functions


To graph y = csc bx or y = sec bx, with b > 0,follow these steps.

Step 1 Graph the corresponding reciprocalfunction as a guide, using a dashedcurve.

y = cos bxy = a sec bxy = a sin bxy = a csc bx

Use as a GuideTo Graph

12

34Rev.S08

Guidelines for Sketching Graphs ofCosecant and Secant Functions Continued


Step 2 Sketch the vertical asymptotes. - They will have equations of the form x = k,

where k is an x-intercept of the graph of the guide function.

Step 3 Sketch the graph of the desired function by drawing the typical U-shapes branches between the adjacent asymptotes. - The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.

35Rev.S08

How to Graph y = 2 sec(x/2)?


Step 1: Graph the corresponding reciprocal functiony = 2 cos (x/2).

The function has amplitude 2 and one period of the graphlies along the interval that satisfies the inequality

Divide the interval into four equal parts.

36Rev.S08

How to Graph y = 2 sec(x/2)? (Cont.)


Step 2: Sketch the vertical asymptotes. These occur at x-values for which the guide function equals 0, such as x =−3π, x = 3π, x = π, x = 3π.

Step 3: Sketch the graph of y = 2 sec x/2 by drawing thetypical U-shaped branches, approaching the asymptotes.

13

37Rev.S08

Tangent Function


38Rev.S08

Cotangent Function


39Rev.S08

Guidelines for Sketching Graphs ofTangent and Cotangent Functions


To graph y = tan bx or y = cot bx, with b > 0,follow these steps.

Step 1 Determine the period, π/b. To locatetwo adjacent vertical asymptotes solvethe following equations for x:

14

40Rev.S08

Guidelines for Sketching Graphs of Tangentand Cotangent Functions continued


Step 2 Sketch the two vertical asymptotes found inStep 1.

Step 3 Divide the interval formed by the verticalasymptotes into four equal parts.

Step 4 Evaluate the function for the first-quarterpoint, midpoint, and third-quarter point, usingthe x-values found in Step 3.

Step 5 Join the points with a smooth curve,approaching the vertical asymptotes. Indicateadditional asymptotes and periods of thegraph as necessary.

41Rev.S08

How to Graph y = tan(2x)?


Step 1: The period of the function is π/2. Thetwo asymptotes have equationsx = −π/4 and x = π/4.

Step 2: Sketch the two vertical asymptotesfound. x = ± π/4.

Step 3: Divide the interval into four equal parts.This gives the following key x-values.

First quarter: −π/8 Middle value: 0 Third quarter: π/8

42Rev.S08

How to Graph y = tan(2x)? (Cont.)


Step 4: Evaluate the function

Step 5: Join the points with asmooth curve, approaching thevertical asymptotes. Indicateadditional asymptotes and periods ofthe graph as necessary.

10−1tan 2x

π/40−π/42x

π/80−π/8x

15

43Rev.S08

How to Graph y = tan(2x)? (Cont.)


Every y value for thisfunction will be 2 unitsmore than thecorresponding y iny = tan x, causing thegraph to be translated 2units up compared toy = tan x.

44Rev.S08

What have we learned?We have learned to:

1. Recognize periodic functions.2. Determine the amplitude and period, when given the equation

of a periodic function.3. Find the phase shift and vertical shift, when given the

equation of a periodic function.4. Graph sine and cosine functions.5. Graph cosecant and secant functions.6. Graph tangent and cotangent functions.7. Interpret a trigonometric model.


45Rev.S08

Credit

Some of these slides have been adapted/modified in part/whole fromthe slides of the following textbook:

• Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8thEdition


Trig m4 Handouts

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