Graphs of Sine & Cosine Functions

MATH 109 - PrecalculusS. 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 com- pared 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:

• 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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bb

cx

b

c 2

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:

a) b)

c) d)

e)24

xy cos3

1

3

2sin2

xy

3cos3 xy

42cos3

2 xy

xy 3sin

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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