G RAPHS OF S INE AND C OSINE FUNCTIONS Objectives: Sketch the graphs of basic sine and cosine functions Use amplitude, period and translations to help.

Slides:



Advertisements
Similar presentations
Graphs of Trigonometric Functions
Advertisements

If you have not watched the PowerPoint on the unit circle you should watch it first. After you’ve watched that PowerPoint you are ready for this one.
Graphs of the Sine, Cosine, & Tangent Functions Objectives: 1. Graph the sine, cosine, & tangent functions. 2. State all the values in the domain of a.
Copyright © Cengage Learning. All rights reserved. 4 Trigonometric Functions.
4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.
Notes Over 6.4 Graph Sine, Cosine Functions Notes Over 6.4 Graph Sine, Cosine, and Tangent Functions Equation of a Sine Function Amplitude Period Complete.
4.3 Vertical and Horizontal Translations
Graphing Sine and Cosine
Graphs of Trigonometric Functions Digital Lesson.
Amplitude, Period, & Phase Shift
Copyright © Cengage Learning. All rights reserved. 4.5 Graphs of Sine and Cosine Functions.
4.5 Sinusoidal Graphs Sketching and Writing Equations.
MAT 204 SP Graphs of the Sine and Cosine Functions 7.8 Phase shift; Sinusoidal Curve Fitting In these sections, we will study the following topics:
1 Properties of Sine and Cosine Functions The Graphs of Trigonometric Functions.
Graphs of Sine and Cosine Five Point Method. 2 Plan for the Day Review Homework –4.5 P odd, all The effects of “b” and “c” together in.
Graphs of Trig Functions
MAT 204 FALL Graphs of the Sine and Cosine Functions 7.8 Phase shift; Sinusoidal Curve Fitting In these sections, we will study the following.
Mathematics Transformation on Trigonometric Functions Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement.
Period and Amplitude Changes
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Symmetry with respect to a point A graph is said to be symmetric with respect to.
2.6 Graphs of the Sine and Cosine Functions xxy = sin x 00=SIN(B2) π/6=PI()/6=SIN(B3) π/3=PI()/3=SIN(B4) π/2=PI()/2=SIN(B5) 2π/3=B5+PI()/6=SIN(B6) 5π/6=B6+PI()/6=SIN(B7)
5.5 a&b Graphs of Sine and Cosine
Section 5.3 Trigonometric Graphs
Amplitude, Period, and Phase Shift
Graphs of Cosine Section 4-5.
4.5 – G RAPHS OF S INE AND C OSINE F UNCTIONS. I N THIS SECTION, YOU WILL LEARN TO :  Sketch the graphs of sine and cosine functions  Use the period.
Section 4.5 Graphs of Sine and Cosine. Overview In this section we first graph y = sin x and y = cos x. Then we graph transformations of sin x and cos.
Chp. 4.5 Graphs of Sine and Cosine Functions p. 323.
Graphs of Trigonometric Functions Digital Lesson.
Graph Trigonometric Functions
Graphs of Trigonometric Functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 DAY 1 : OBJECTIVES 1. Define periodic function.
Chapter 14 Day 8 Graphing Sin and Cos. A periodic function is a function whose output values repeat at regular intervals. Such a function is said to have.
4.5 Graphs of Trigonometric Functions 2014 Digital Lesson.
Graphs of Trigonometric Functions Digital Lesson.
Graphs of Trigonometric Functions. Properties of Sine and Cosine Functions 2 6. The cycle repeats itself indefinitely in both directions of the x-axis.
Copyright © 2007 Pearson Education, Inc. Slide Graphs of the Sine and Cosine Functions Many things in daily life repeat with a predictable pattern.
Sections 7.6 and 7.8 Graphs of Sine and Cosine Phase Shift.
1 Properties of Sine and Cosine Functions MATH 130 Lecture on The Graphs of Trigonometric Functions.
Precalculus 1/9/2015 DO NOW/Bellwork: 1) Take a unit circle quiz 2) You have 10 minutes to complete AGENDA Unit circle quiz Sin and Cosine Transformations.
5.1 Graphing Sine and Cosine Functions
Unit 7: Trigonometric Functions Graphing the Trigonometric Function.
Graphs of Sine and Cosine
Graphs of Cosine Functions (part 2)
Properties of Sine and Cosine Functions
Trigonometric Graphs 6.2.
Amplitude, Period, & Phase Shift
2.7 Sinusoidal Graphs; Curve Fitting
Graphs of Trigonometric Functions
Graphs of Sine and Cosine Functions
Graphs of Trigonometric Functions
Graphs of Trigonometric Functions
Graphs of Trigonometric Functions
Trigonometric Graphs 1.6 Day 1.
Graphs of Trigonometric Functions
Amplitude, Period, and Phase Shift
Graphs of Trigonometric Functions
Trigonometric Functions
Amplitude, Period, & Phase Shift
Unit 7: Trigonometric Functions
Graphs of Trigonometric Functions
Copyright © Cengage Learning. All rights reserved.
Graphs of Trigonometric Functions
Graphs of Trigonometric Functions
4.4 Graphs of Sine and Cosine Functions
Graphs of Trigonometric Functions
5.1 Graphing Sine and Cosine Functions
Graphs of Trigonometric Functions
Graphs of Sine and Cosine Functions
7.3 Periodic Graphs & Amplitude Objectives:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Presentation transcript:

G RAPHS OF S INE AND C OSINE FUNCTIONS Objectives: Sketch the graphs of basic sine and cosine functions Use amplitude, period and translations to help sketch the graphs of sine and cosine functions

G RAPHING S INE AND C OSINE F UNCTIONS In this unit, we will take the unit circle introduced in the previous unit, and graph it on the Cartesian plane. To do this, we are going to “unravel” the unit circle. Recall that for the unit circle the coordinates are where is the central angle. To graph, rewrite the coordinates as where is the central angle, in radians. We expanded the sine coordinates for.

S INE CURVE

Notice that the curve ranges from 1 to -1. The maximum value is 1, which is at. The minimum value is -1 at. This “height” of the sine function is called the amplitude. If we had continued the curve, it would repeat. This means that the sine curve is periodic. Look back at the unit circle, the sine value changes until it reaches. Therefore, the curve will repeat every units, making the period. The domain is all real numbers.

C OSINE CURVE Similarly, when we expand the cosine curve, from the unit circle, we have: Notice that the range is also and the domain is also. The period is also

B ASIC S INE AND C OSINE C URVES o Comparing, we see that the curves are almost identical, except that the sine curve starts at and the cosine curve starts at.

C HARACTERISTICS OF S INE AND C OSINE One Cycle: the period of each of these curves is from. The curves repeat indefinitely to the left and right. Domain: Range: Symmetry: Sine curve is symmetric with respect to the origin (odd function) and Cosine curve is symmetric with respect to the y-axis (even function)

C HARACTERISTICS OF S INE AND C OSINE To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points and minimum points. Sine key points: Cosine key points:

T RANSFORMATIONS OF SINE AND COSINE You will study the graphic effect of each of the constants in equations of the forms: Sine : Cosine: Amplitude represents half the distance between the max and the min values of the function (multiply the y-values) If, the basic curve is stretched and if is shrunk Period the horizontal distance needed to complete one cycle of the function If, the period of the graph is greater than and if is less than

T RANSFORMATIONS OF SINE AND COSINE Phase shift : the c amount shifted right or left to determine the new endpoints of the one cycle If, the graph has a horizontal translation to the right and if to the left. Vertical translation: the d amount shifted up or down to determine the new max and min of the one cycle If, the graph has a vertical translation up and if the graph goes down Reflection: the sign in front of the determines if it is reflected over the x-axis

EX: F IND THE AMPLITUDE, THE PERIOD, THE REFLECTION, ANY VERTICAL TRANSLATION, AND ANY PHASE SHIFT OF EACH GRAPH

EX: S KETCH THE G RAPH OF ONE CYCLE 4.

EX: S KETCH THE G RAPH OF ONE CYCLE 5.

EX: S KETCH THE G RAPH OF ONE CYCLE 6.

EX: S KETCH THE G RAPH OF ONE CYCLE 7.