Sect 4 5 Graphing Trig Functions Sine and

Take a look at the graph of y = sin x: (one cycle) Some

Using Key Points to Graph the Sine Curve Once you know the basic shape

Take a look at the graph of y = cos x: (one cycle) Some

Using Key Points to Graph the Cosine Curve Once you know the basic shape

Characteristics of the Graphs of y = sin x and y = cos x

Transformations of the graphs of the trig functions n Reflections over x-axis n Vertical

II. Vertical Stretching or Compression (Amplitude change) Example 10

III. Horizontal Stretching or Shrinking/Compression (Period change) Example 11

Examples State the amplitude and period for each function. Then graph one cycle of

Example: Determine the amplitude, period, and phase shift of the function. Then sketch the

Example: List all of the transformations that the graph of y = sin x

Slides: 19

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Sect. 4 -5 Graphing Trig Functions Sine and Cosine 1

The Unit Circle

Take a look at the graph of y = sin x: (one cycle) Some points on the graph: 3

Using Key Points to Graph the Sine Curve Once you know the basic shape of the sine curve, you can use the key points to graph the sine curve by hand. The five key points in each cycle (one period) of the graph are the intercepts, the maximum point, and the minimum point. 4

Take a look at the graph of y = cos x: (one cycle) Some points on the graph: 5

Using Key Points to Graph the Cosine Curve Once you know the basic shape of the cosine curve, you can use the key points to graph the cosine curve by hand. 6

Characteristics of the Graphs of y = sin x and y = cos x n. Domain: n. Range: _____________ n. Amplitude: The amplitude of the sine and cosine functions is half the distance between the maximum and minimum values of the function. The amplitude of both y= sin x and y = cos x is _______. n. Period: The length of the interval needed to complete one cycle. The period of both y= sin x and y = cos x is ____. 7

Transformations of the graphs of the trig functions n Reflections over x-axis n Vertical Stretches or Shrinks n Horizontal Stretches or Shrinks/Compression n Vertical Shifts n Phase shifts (Horizontal shifts/displacement) 8

I. Reflections over x-axis Example: 9

II. Vertical Stretching or Compression (Amplitude change) Example 10

III. Horizontal Stretching or Shrinking/Compression (Period change) Example 11

y x 12

Examples State the amplitude and period for each function. Then graph one cycle of each function by hand. Verify using your graphing calculator. y x 13

y x 14

IV. Phase Shifts Example 15

IV. Phase Shifts (continued) Example 16

Example: Determine the amplitude, period, and phase shift of the function. Then sketch the graph of the function by hand. y x 17

Example: List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand. y x 18

Example: List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand. y x 19