Graphs of Sine

1
Graphs of Sine Cosine Functions

• MATH 109 - Precalculus
• S. Rook

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Overview

• Section 4.5 in the textbook
• Graphs of parent sine cosine functions
• Transformations of sine cosine graphs affecting
  the y-axis
• Transformations of sine cosine graphs affecting
  the x-axis
• Graphing y d a sin(bx c) or
  y d a cos(bx c)

3
Graphs of Parent Sine Cosine Functions
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Graph of the Parent Sine Function y sin x

• Recall that on the unit circle any point (x, y)
  can be written as (cos ?, sin ?)
• Also recall that the period of y sin x is 2p
• Thus, by taking the
  
  y-coordinate of each
  point on the
  
  circumference of the
  unit circle
  we generate
  one cycle of y sin
  x,
  0 lt x lt 2p

5
Graph of the Parent Sine Function y sin x
(Continued)

• To graph any sine function we need to know
• A set of points on the parent function y sin x
• (0, 0), (p/2, 1), (p, 0), (3p/2, -1), (2p, 0)
• Naturally these are not the only points, but are
  often the easiest to manipulate
• The shape of the graph

6
Graph of the Parent Cosine Function y cos x

• Recall that on the unit circle any point (x, y)
  can be written as (cos ?, sin ?)
• Also recall that the period of y cos x is 2p
• Thus, by taking the
  
  x-coordinate of each
  point on the
  
  circumference of the
  unit circle
  we generate
  one cycle of y cos
  x,
  0 lt x lt 2p

7
Graph of the Parent Cosine Function y cos x
(Continued)

• To graph any cosine function we need to know
• A set of points on the parent function y cos x
• (0, 1), (p/2, 0), (p, -1), (3p/2, 0), (2p, 1)
• Naturally these are not the only points, but are
  often the easiest to manipulate
• The shape of the graph

8
Transformations of Sine Cosine Graphs Affecting
the y-axis
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Transformations of Sine Cosine Graphs

• The graph of a sine or cosine function can be
  affected by up to four types of transformations
• Can be further classified as affecting either the
  x-axis or y-axis
• Transformations affecting the x-axis
• Period
• Phase shift
• Transformations affecting the y-axis
• Amplitude
• Reflection
• Vertical translation

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Amplitude

• Amplitude is a measure of the distance between
  the midpoint of a sine or cosine graph and its
  maximum or minimum point
• Because amplitude is a distance, it MUST be
  positive
• Can be calculated by averaging the minimum and
  maximum values (y-coordinates)
• Thus ONLY functions with a minimum AND maximum
  point can possess an amplitude
• Represented as a constant a being multiplied
  outside of y sin x or y cos x
• i.e. y a sin x or y a cos x

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How Amplitude Affects a Graph

• Amplitude constitutes a vertical stretch
• Multiply each y-coordinate by a
• If a gt 1
• The graph is stretched
  in the
  y-direction in
  comparison to the
  
  parent graph
• If 0 lt a lt 1
• The graph is
  compressed in the
  
  y-direction in
  
  comparison to the
  parent
  graph

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How Amplitude Affects a Graph (Continued)

• Recall that the range of y sin x and y cos x
  is -1, 1
• Thus the range of y a sin x and y a cos x
  becomes -a, a

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How Reflection Affects a Graph

• Reflection occurs when a lt 0
• Reflects the graph over the y-axis

14
How Vertical Translation Affects a Graph

• Vertical Translation constitutes a vertical shift
• Add d to each y-coordinate
• If d gt 0
• The graph is shifted
  up by d
  units in
  comparison to the
  
  parent graph
• If d lt 0
• The graph is shifted
  down by
  d units in
  comparison to the
  
  parent graph

15
Transformations of Sine Cosine Graphs Affecting
the x-axis
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How Phase Shift Affects a Graph

• Phase shift constitutes a horizontal shift
• Add -c to each x-coordinate (the opposite value!)
• If c is inside
• The graph shifts to the
  left c
  units when
  compared to the parent
  
  graph
• If -c is inside
• The graph shifts to the
  right c units
  when
  compared to the parent
  graph

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Period

• Recall that informally the period is the length
  required for a function or graph to complete one
  cycle of values
• Represented as a constant b multiplying the x
  inside the sine or cosine
• i.e. y sin(bx) or y cos(bx)

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How Period Affects a Graph

• Changes in the period are horizontal shifts
• Multiply each x-coordinate by 1/b
• If b gt 1
• The graph is compressed
  
  resulting in more cycles in
  the
  interval 0 to 2p as com-
  pared with the
  parent graph
• If 0 lt b lt 1
• The graph is stretched
  resulting in less
  cycles in
  the interval 0 to 2p as
  
  compared with the parent
  graph

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Graphing y d a sin(bx c) or y d a
cos(bx c)
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Establishing the y-axis

• The key to graphing either y d a sin(bx c)
  or y d a cos(bx c) is to establish
  the graph skeleton
• i.e. how the x-axis and y-axis will be marked
• Establish the y-axis
• Determined by amplitude and vertical translation
• Find a and d
• Range for parent -1 y 1
• After factoring in amplitude -a y a
• After factoring in vertical translation
  -a d y
  a d

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Establishing the x-axis

• Establish the x-axis (two methods)
• Method I Interval method
• Solve the linear inequality 0 bx c 2p for x
• Generally
• Left end of the interval is where one cycle
  starts (phase shift)
• Right end of the interval is where one cycle ends
• Period is obtained by subtracting the two
  endpoints (right left)

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Establishing the x-axis (Continued)

• Method II Formulas
• P.S. -c/b
• P 2p/b
• End of a cycle occurs at P.S. P
• Divide the period into 4 equal subintervals to
  get a step size
• Starting with the phase shift, continue to apply
  the step size until the end of the cycle is
  reached
• These 5 points correlate to the 5 original points
  for the parent graph

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Graphing y d a sin(bx c) or y d a
cos(bx c)

• To graph y d a sin(bx c) or y d a
  cos(bx c)
• Establish the y-axis
• Establish the x-axis
• The x-values of the 5 points in the are the
  transformed x-values for the final graph
• Use transformations to calculate the y-values for
  the final graph
• Connect the points in a smooth curve in the shape
  of a sine or cosine this is 1 cycle
• Be aware of reflection when it exists
• Extend the graph if necessary

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Graphing y d a sin(bx c) or y d a
cos(bx c) (Example)

• Ex 1 Graph by finding the amplitude, vertical
  translation, phase shift, and period include 1
  additional full period forwards and ½ a period
  backwards
• a) b)
• c) d)
• e)

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Summary

• After studying these slides, you should be able
  to
• Understand the shape and selection of points that
  comprise the parent cosine and sine functions
• Understand the transformations that affect the
  y-axis
• Understand the transformations that affect the
  x-axis
• Graph any sine or cosine function
• Additional Practice
• See the list of suggested problems for 4.5
• Next lesson
• Graphs of Other Trigonometric Functions (Section
  4.6)