Presentation on theme: "4-5 graphs of sine and cosine functions"— Presentation transcript:
1 4-5 graphs of sine and cosine functions Chapter 44-5 graphs of sine and cosine functions
2 Objectives Sketch the graphs of basic sine and cosine functions. • Use amplitude and period to help sketch the graphs of sine and cosine functions.• Sketch translations of the graphs of sine and cosine functions.• Use sine and cosine functions to model real-life data.
3 Basic sine and cosine functions The Sine Curve y = a sin tThe sine curve occurs naturally when we are examining waves. When waves have more energy, they go up and down more vigorously. We say they have greater amplitude.Let's investigate the shape of the curve y = a sin t and see what the concept of "amplitude" means.
4 Basic sine and cosine functions The black portion of the graph represents one period of the function and is called one cycle of the sine curve. The domain of the sine and cosine functions is the set of all real numbers. The range of each function is the interval [–1, 1]. Each function has a period of 2 .
5 AmplitudeThe "a" in the expression y = a sin x represents the amplitude of the graph. It is an indication of how much energy the wave contains. The amplitude is the distance from the "resting" position (otherwise known as the mean value or average value) of the curve. In the interactive above, the amplitude can be varied from 10 to 100 units. Amplitude is always a positive quantity. We could write this using absolute value signs. For the curve y = a sin x, amplitude =|a|
6 Period The variable b in both of the graph types y = a sin bx y = a cos bxaffects the period (or wavelength) of the graph. The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again.The relationship between b and the period is given by:Period=2π/ b Note: As b gets larger, the period decreases.
7 Phase shiftsy= a sin(bx + c) and y = a cos(bx + c) Both b and c in these graphs affect the phase shift (or displacement), given by: Phase shift=−c/ b The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx+c=0 .
8 Keys to graph sine and cosine functions Five key points in one period of each graph: the intercepts, maximum points, and minimum points
9 Example#1Find the period and amplitude of the function y = 10 sin 5x .
10 Example#2What is the period and amplitude of the function y = cos 12x ?
11 Example#3What is the period and amplitude of the function y = (- 5/ 6) sin 3x?
12 Student guided practice Do problems in your book page 299
13 Example Sketch the graph of y = 2 sin x on the interval [– , 4 ]. Solution: Note that y = 2 sin x indicates that the y-values for the key points will have twice the magnitude of those on the graph of y = sin x.Divide the period 2 into four equal parts to get the key points for y = 2 sin x.Intercept Maximum Intercept Minimum Interceptand= 2(sin x)7Example 1 – SolutionBy connecting these key points with a smooth curve and extending the curve in both directions over the interval [– , 4 ], you obtain the graph shown in Figure 4.50.c
15 Example Sketch the graph of 𝑦=sin( 𝑥 2 ) Solution: The amplitude is 1. Moreover, because b = 1/2 , the period is 4𝜋.Now, divide the period-interval [0, 4 ] into four equal partswith the values , 2 , and 3 to obtain the key points on thegraph.Intercept Maximum Intercept Minimum Intercept (0, 0), ( 𝜋, 1), (2 , 0), (3 , –1), and (4 , 0)