MAC 1114

1
MAC 1114

• Module 4
• Graphs of the Circular Functions

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Learning Objectives

• Upon completing this module, you should be able
  to
• Recognize periodic functions.
• Determine the amplitude and period, when given
  the equation of a periodic function.
• Find the phase shift and vertical shift, when
  given the equation of a periodic function.
• Graph sine and cosine functions.
• Graph cosecant and secant functions.
• Graph tangent and cotangent functions.
• Interpret a trigonometric model.

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3
Graphs of the Circular Functions
There are three major topics in this module
- Graphs of the Sine and Cosine Functions -
Translations of the Graphs of the Sine and Cosine
Functions - Graphs of the Other Circular Functions
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4
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,
  every integer n, and some positive real number p.
  The smallest possible positive value of p is the
  period of the function.

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5
Example of a Periodic Function
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Example of Another Periodic Function
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What is the Amplitude of a Periodic Function?

• The amplitude of a periodic function is half the
  difference between the maximum and minimum
  values.
• The graph of y a sin x or y a cos x, with
  a ? 0, will have the same shape as the graph of
  y sin x or y cos x, respectively, except the
  range will be -a, a. The amplitude is a.

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How to Graph y 3 sin(x) ?
Note the difference between sin x and 3sin x.
What is the difference?
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How to Graph y sin(2x)?

• The period is 2p/2 p. The graph will complete
  one period over the interval 0, p.
• The endpoints are 0 and p, the three middle
  points are
• Plot points and join in a smooth curve.

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10
How to Graph y sin(2x)?(Cont.)
Note the difference between sin x and sin 2x.
What is the difference?
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Period of a Periodic Function

• Based on the previous example, we can generalize
  the following
• For b gt 0, the graph of y sin bx will resemble
  that of y sin x, but with period 2p/b.
• The graph of y cos bx will resemble that of
  y cos x, with period 2p/b.

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

• The period is 3p.
• Divide the interval into four equal parts.
• Obtain key points for one period.

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

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

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Guidelines for Sketching Graphs of Sine and
Cosine Functions

• To graph y a sin bx or y a cos bx, with
  b gt 0, follow these steps.
• Step 1 Find the period, 2p/b. Start with 0 on
  the x-axis, and lay off a distance of 2p/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.

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Guidelines for Sketching Graphs of Sine and
Cosine Functions Continued

• 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,
  to the right and to the left, as needed.

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How to Graph y -2 sin(4x)?

• Step 1 Period 2p/4 p/2. The function will
  be graphed over the interval 0, p/2 .
• Step 2 Divide the interval into four equal parts.
• Step 3 Make a table of values

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How to Graph y -2 sin(4x)? (Cont.)

• Plot the points and join them with a sinusoidal
  curve with amplitude 2.

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What is a Phase Shift?

• In trigonometric functions, a horizontal
  translation is called a phase shift.
• In the equation
• the graph is shifted p/2 units to the right.

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How to Graph y sin (x - p/3) by Using
Horizontal Translation or Phase Shift?

• Find the interval for one period.
• Divide the interval into four equal parts.

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How to Graph y sin (x - p/3) by Using
Horizontal Translation or Phase Shift?(Cont.)
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How to Graph y 3 cos(x p/4) by Using
Horizontal Translation or Phase Shift?

• Find the interval.
• Divide into four equal parts.

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How to Graph y 3 cos(x p/4) by Using
Horizontal Translation or Phase Shift?
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23
How to Graph y 2 - 2 sin 3x by Using Vertical
Translation or Vertical Shift?

• The graph is translated 2 units up from the graph
  y -2 sin 3x.

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How to Graph y 2 - 2 sin 3x by Using Vertical
Translation or Vertical Shift?(Cont.)

• Plot the points and connect.

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Further Guidelines for Sketching Graphs of Sine
and Cosine Functions

• Method 1 Follow these steps.
• Step 1 Find an interval whose length is one
  period 2p/b by solving the three part
  inequality 0 b(x - d) 2p.
• 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 (middle points of the wave.)

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

• 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,
  to the right and to the left, as needed.
• Method 2 First graph the basic circular
  function. The amplitude of the function is a,
  and the period is 2p/b. Then use translations
  to graph the desired function. The vertical
  translation is c units up if c gt 0 and c
  units down if c lt 0. The horizontal translation
  (phase shift) is d units to the right if d gt 0
  and d units to the left if d lt 0.

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How to Graph y -1 2 sin (4x p)?

• Step 2 Divide the interval.
• Step 3 Table

• Write the expression in the form c a sin b(x -
  d) by rewriting 4x p as
• Step 1

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How to Graph y -1 2 sin (4x p)?(Cont.)
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How to Graph y -1 2 sin (4x p)?(Cont.)

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

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Lets Take a Look at Other Circular Functions.
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Cosecant Function
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Secant Function
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Guidelines for Sketching Graphs of Cosecant and
Secant Functions

• To graph y csc bx or y sec bx, with b gt 0,
  follow these steps.
• Step 1 Graph the corresponding reciprocal
  function as a guide, using a dashed curve.

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Guidelines for Sketching Graphs of Cosecant 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.

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How to Graph y 2 sec(x/2)?
Step 1 Graph the corresponding reciprocal
function y 2 cos (x/2). The function has
amplitude 2 and one period of the graph lies
along the interval that satisfies the
inequality Divide the interval into four equal
parts.
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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 -3p, x 3p, x p, x 3p.
• Step 3 Sketch the graph of y 2 sec x/2 by
  drawing the typical U-shaped branches,
  approaching the asymptotes.

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Tangent Function
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Cotangent Function
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Guidelines for Sketching Graphs of Tangent and
Cotangent Functions

• To graph y tan bx or y cot bx, with b gt 0,
  follow these steps.
• Step 1 Determine the period, p/b. To locate
  two adjacent vertical asymptotes solve the
  following equations for x

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Guidelines for Sketching Graphs of Tangent and
Cotangent Functions continued

• Step 2 Sketch the two vertical asymptotes found
  in Step 1.
• Step 3 Divide the interval formed by the vertical
  asymptotes into four equal parts.
• Step 4 Evaluate the function for the
  first-quarter point, midpoint, and
  third-quarter point, using the x-values found
  in Step 3.
• Step 5 Join the points with a smooth curve,
  approaching the vertical asymptotes. Indicate
  additional asymptotes and periods of the
  graph as necessary.

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How to Graph y tan(2x)?

• Step 1 The period of the function is p/2. The
  two asymptotes have equations x
  -p/4 and x p/4.
• Step 2 Sketch the two vertical asymptotes
  found. x p/4.
• Step 3 Divide the interval into four equal
  parts. This gives the following key x-values.
• First quarter -p/8
• Middle value 0 Third quarter p/8

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

• Step 4 Evaluate the function
• Step 5 Join the points with a smooth curve,
  approaching the vertical asymptotes. Indicate
  additional asymptotes and periods of the graph
  as necessary.

p/8
0
-p/8
x
p/4
0
-p/4
2x
1
0
-1
tan 2x
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How to Graph y tan(2x)? (Cont.)

• Every y value for this function will be 2 units
  more than the corresponding y in y tan x,
  causing the graph to be translated 2 units up
  compared to y tan x.

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What have we learned?

• We have learned to
• Recognize periodic functions.
• Determine the amplitude and period, when given
  the equation of a periodic function.
• Find the phase shift and vertical shift, when
  given the equation of a periodic function.
• Graph sine and cosine functions.
• Graph cosecant and secant functions.
• Graph tangent and cotangent functions.
• Interpret a trigonometric model.

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to download other modules.
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45
Credit

• Some of these slides have been adapted/modified
  in part/whole from the slides of the following
  textbook
• Margaret L. Lial, John Hornsby, David I.
  Schneider, Trigonometry, 8th Edition

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Rev.S08