13 Trigonometric Graphs

Transcription

1 Trigonometric Graphs Concepts: Period The Graph of the sin, cos, tan, csc, sec, and cot Functions Appling Graph Transformations to the Graphs of the sin, cos, tan, csc, sec, and cot Functions Using Graphical Evidence to Make Conjectures about Identities The Periodicit Identities The Relationship Between the Periodicit Identities and the Periods of the Trigonometric Graphs (Sections 6. and 6.5). Period Intuitivel, the graph of a function with period k is a graph that does not change when ou shift the graph to the right or left k units. More formall, Definition. The graph of f() is periodic with period k if f(+k) = f() and Eample. Find the period the following graph f( k) = f()

2 . The Graph of = sin() Recall the unit circle definition for sin(). Let s see how we can use the unit circle to sketch the graph of = sin(). (a,b) θ The Graph of = sin(): (θ,b) - Corresponding Periodicit Identit:

3 Eample. List the transformations needed to convert the graph of = sin() into the graph of = sin(+ ). What is the period of the graph? Sketch the graph. Eample. List the transformations needed to convert the graph of = sin() into the graph of = sin( ). What is the period of the graph? Sketch the graph. Eample.5 List the transformations needed to convert the graph of = sin() into the graph of = Asin(b+c). What is the period of the graph?

4 . The graph of = cos() (a,b) θ The Graph of = cos(): (θ,a) - Corresponding Periodicit Identit:

5 Eample.6 List the transformations needed to convert the graph of = cos() into the graph of = cos( + ). What is the period of the graph? Sketch the graph. Eample.7 Use graphical evidence to determine which of the following MIGHT be trigonometric identities and which definitel cannot be trigonometric identities. sin(+ ) = cos() sin() = sin() cos() Eample.8 In the eample above, we saw that sin() sin()cos(). Can ou modif the original equation to find a new equation that has the possibilit of being an identit? Eample.9 This eample is taken from Eercise 6 in Section 6. of our tetbook. If ou have a TI-8 or TI-8 calculator, look at the graphs of = cos() and = in the viewing window [,88] [,]. Trace the graph of = cos(). Wh do ou think the calculator displas the wrong graph? What are ou supposed to learn from this eample? 5

6 . The graph of = tan() Let be an angle in standard position. Then the terminal side of that angle intersects the unit circle at the point. What is the slope of the terminal side of the angle? To understand the graph of the tangent function, it helps to think of the terminal side of an angle sweeping around the unit circle. 5 (a,b) ( θ, b a ) θ Corresponding Periodicit Identit: 6

7 .5 The graph of = csc() Use the graph of = sin() and the definition of the csc function to sketch the graph of = csc() Corresponding Periodicit Identit: Eample. Sketch the graph of = csc(). 7

8 .6 The graph of = sec() Use the graph of = cos() and the definition of the sec function to sketch the graph of = sec() Corresponding Periodicit Identit: Eample. Sketch the graph of = csc(+). 8

9 .7 The graph of = cot() Use the graph of = tan() and a reciprocal identit to sketch the graph of = cot() Corresponding Periodicit Identit: 9

10 .8 An Application Eample. (Eample 9 from Section 6.5 of our tetbook) Kat s blood pressure can be modeled b the function f(t) = cos(.5t)+95, where t is the time (in seconds) and f(t) is in millimeters of mercur. The highest pressure (sstolic) occurs when the heart beats, and the lowest pressure (diastolic) occurs when the heart is at rest between beats. The blood pressure is the ratio sstolic/diastolic. (a) Graph the blood pressure function over a period of two seconds and determine Kat s blood pressure. (b) Find Kat s pulse rate (number of heartbeats per minute). Eample. (a) Jake has a blood pressure of /8. Assume that Jake s heart beats once ever second. Find a model for Jake s blood pressure. (b) Joni has a blood pressure of /7. Assume that Joni s heart rate is 8 beats per minute. Find a model for Joni s blood pressure.