Amplitude, Reflection, and Period

TrigonometryMATH 103

S. Rook

Overview

• Section 4.2 in the textbook:– Amplitude and Reflection– Period– Graphing y = A sin Bx or y = A cos Bx

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Amplitude and Reflection

Amplitude

• If a given graph has both a minimum value m AND a maximum value M, then the amplitude is– Only the sine and cosine graphs

possess this property– The minimum and maximum

value for both y = cos x and y = sin x is -1 and 1 respectively

– Thus the amplitude for y = sin x and y = cos x is

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mM 2

1

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111

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Range of the Sine and Cosine Functions

• Recall that the range is the allowable set of y-values for a function– We just observed that the minimum value is -1

and the maximum value is 1 for y = sin x and y = cos x

i.e. -1 ≤ y ≤ 1

• For both y = sin x and y = cos x:– Domain: (-oo, +oo)– Range: [-1, 1]

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How Amplitude Affects a Graph

• The graphs of y = A sin x and y = A cos x are related to the graphs y = sin x and y = cos x:– Each y-coordinate of y = sin x or y = cos x is multiplied by A

to get the new functions y = A sin x or y = A cos x• E.g. (0, 1) on y = cos x would become (0, 5) on the graph

of y = 5 cos x

• Amplitude = |A|– Always positive– The maximum value is |A| and the minimum value is -|

A|– The range of y = A sin x or y = A cos x is then [-|A|, |A|]

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How Amplitude Affects a Graph (Continued)

• If 0 < A < 1y = A sin x or y = A cos x will be COMPRESSED in the y-direction as compared to y = sin x or y = cos x

• If A > 1y = A sin x or y = A cos x will be STRETCHED in the y-direction as compared to y = sin x or y = cos x

• The value of A affects ONLY the y-coordinate• The value of A does NOT affect the period– e.g. y = sin x and y = 4 sin x both have period 2π

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Graphing y = A sin x or y = A cos x

• To graph one cycle of y = A sin x or y = A cos x:– Divide the interval from 0 to 2π into 4 equal subintervals:

• The x-axis will be marked by increments of π⁄2

• The y-axis will have a minimum value of -|A| and a maximum value of |A|

• We can use so few points because we know the shape of the sine or cosine graph!

– Create a table of values • Based on the values labeled on the x-axis

– Connect the points to make the graph• Based on the shape of either the sine or cosine graph

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Amplitude (Example)

Ex 1: Sketch one complete cycle:

a) y = 3⁄4 sin x

b) y = 5 cos x

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Reflection

• If A < 0 y = A sin x or y = A cos x will be reflected about the x-axis• Recall that multiplying the y-coordinate of a point by a

negative value reflects the point over the x-axis– E.g. (3, 2) reflected over the x-axis becomes (3, -2)

Amplitude = |A|• Maximum value is still |A| and minimum value is still -|A|

• Repeat the EXACT same steps to graph y = A sin x or y = A cos x when A < 0

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Reflection (Example)

Ex 2: Sketch one complete cycle:y = -3 cos x

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Period

Introduction to How the Argument Affects the Period

• Recall that informally the period is the smallest interval until the graph starts to repeat– The period of both y = sin x and y = cos x is 2π

• Now we will consider the effects of multiplying the argument (input) by a constant B– i.e. How is y = sin Bx or y = cos Bx different from

y = sin x or y = cos x?– Note that in the case of y = sin x or y = cos x, B = 1

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How Period Affects a Graph

• Consider graphing y = sin x, y = sin 2x, and y = sin 4x using a table of values– Notice that, on the interval 0 to 2π, y = sin x makes 1

cycle, y = sin 2x makes 2 cycles, and y = sin 4x makes 4 cycles

– The period of y = sin x is 2π, the period of y = sin 2x is π, and the period of y = sin 4x is π⁄2

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How Period Affects a Graph (Continued)

• To establish a relationship between y = sin x and y = sin Bx or y = cos x and y = cos Bx:– When B = 1, the graph makes 1 cycle in the interval 0 to 2π

and the period is 2π

– When B = 2, the graph makes 2 cycles in the interval 0 to 2π and the period is π

(divide by 2)– When B = 4, the graph makes 4 cycles in the interval 0 to

2π and the period is π⁄2

(divide by 4)15

20 x

xx 0220

20240

xx

Relationship Between B and Period

• Therefore, for y = sin Bx or y = cos Bx:

• To graph one cycle, we repeat the same steps for graphing y = A sin x or y = A cos x EXCEPT:– The period may NOT necessarily be 2π– Divide the interval between 0 and the period into 4 equal

subintervals• 4 is not a “magic number” but an easy number to utilize in the

calculations – will always get 0, π⁄2, π, 3π⁄2, 2π

• The value of B affects ONLY the x-coordinate• The value of B does NOT affect the amplitude

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BBxBx

2Period

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Period (Example)

Ex 3: Sketch one complete cycle:y = cos 2x

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Graphing y = A sin Bx or y = A cos Bx

Graphing y = A sin Bx or y = A cos Bx

• Given y = A sin Bx or y = A cos Bx:|A| is the amplitude is the period

• To graph y = A sin Bx or y = A cos Bx:– Calculate the amplitude and period– Graph one cycle by dividing the interval from 0 to the period

into 4 equal subintervals• We will discuss intervals OTHER THAN 0 to the period when we

discuss phase shift in the next lesson• Textbook refers to this as “Constructing a Frame”

– Extend the graph as necessary

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B

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Graphing y = A sin Bx or y = A cos Bx (Example)

Ex 4: Graph over the given interval:

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43 ,sin2 xxy

Graphing y = A sin Bx or y = A cos Bx (Example)

Ex 5: Give the amplitude and period of the graph:

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Summary

• After studying these slides, you should be able to:– Graph a sine or cosine function for any amplitude and

period– Identify the amplitude and period of a sine or cosine

graph

• Additional Practice– See the list of suggested problems for 4.2

• Next lesson– Vertical Translation and Phase Shift (Section 4.3)

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