Amplitude, Reflection, and Period Trigonometry MATH 103 S. Rook

Overview Section 4.2 in the textbook: – Amplitude and Reflection – Period – Graphing y = A sin Bx or y = A cos Bx 2

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 4

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] 5

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|] 6

How Amplitude Affects a Graph (Continued) If 0 < A < 1 y = 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 > 1 y = 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 7

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 8

Amplitude (Example) Ex 1: Sketch one complete cycle: a) y = 3 ⁄ 4 sin x b) y = 5 cos x 9

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 10

Reflection (Example) Ex 2: Sketch one complete cycle: y = -3 cos x 11

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 13

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 14

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

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 16

Period (Example) Ex 3: Sketch one complete cycle: y = cos 2x 17

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 19

Graphing y = A sin Bx or y = A cos Bx (Example) Ex 4: Graph over the given interval: 20

Graphing y = A sin Bx or y = A cos Bx (Example) Ex 5: Give the amplitude and period of the graph: 21

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) 22