Transcript Angles, Degrees, and Special Triangles

Graphs of Sine & Cosine
Functions
MATH 109 - Precalculus
S. Rook
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)
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Graphs of Parent Sine & Cosine
Functions
Graph of the Parent Sine Function
y = sin x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• Also recall that the period of y = sin x is 2π
• 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 < x < 2π
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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), (π⁄2, 1), (π, 0), (3π⁄2, -1), (2π, 0)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
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Graph of the Parent Cosine
Function y = cos x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• Also recall that the period of y = cos x is 2π
• 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 < x < 2π
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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), (π⁄2, 0), (π, -1), (3π⁄2, 0), (2π, 1)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
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Transformations of Sine & Cosine
Graphs Affecting the y-axis
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 > 1
• The graph is stretched
in the y-direction in
comparison to the
parent graph
– If 0 < a < 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 < 0
– Reflects the graph over the y-axis
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How Vertical Translation Affects a
Graph
• Vertical Translation constitutes a vertical shift
– Add d to each y-coordinate
– If d > 0
• The graph is shifted
up by d units in
comparison to the
parent graph
– If d < 0
• The graph is shifted
down by d units in
comparison to the
parent graph
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Transformations of Sine & Cosine
Graphs Affecting the x-axis
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 > 1
• The graph is compressed
resulting in more cycles in
the interval 0 to 2π as compared with the parent graph
– If 0 < b < 1
• The graph is stretched
resulting in less cycles in
the interval 0 to 2π as
compared with the parent
graph
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Graphing y = d + a sin(bx + c) or
y = d + a cos(bx + c)
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 ≤ 2π for x
– Generally:
c
c 2
 x 
b
b b
• 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 = 2π⁄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:
1
y

a) 3 cos x
c)
y  2  sin
b)
2x
3
2
x 
e) y  3 cos 2  4 


y  sin3x   
d) y  3 cosx     3
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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)
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