Section 2.4 General Sinusoidal Graphs

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Marshall Pitts
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1 Section. General Graphs Objective: any one of the following sets of information about a sinusoid, find the other two: ) the equation ) the graph 3) the amplitude, period or frequency, phase displacement, and vertical displacement In this section, you will put together the ideas from Sections. and.3 to graph functions whose equations have the y VD A sin ( PD ) y VD A cos ( PD ). form or The four constants A,, VD, and PD have the following effects: ) A is the amplitude. The absolute value is needed since the constant A may be a negative number. So, Amplitude A. ) is the number of cycles the sinusoid makes in, so the period P. 3) VD is the vertical displacement. ) PD is the phase displacement. If is negative, use the absolute value of in the period calculation. Your text s problems do not have negative values, but you may run into negative values in the future. The period is the number of degrees per cycle. It is sometimes convenient to speak of the number of cycles per degree. This quantity is called the frequency. The frequency of a periodic function is the reciprocal of the period. So, frequency and period. period frequency An efficient stepwise procedure for drawing a sinusoid is: ) Draw the sinusoidal axis. ) Draw upper and lower bounds by going A units above and below the sinusoidal axis. 3) Find the starting point of a cycle at PD, the phase displacement. Cosine functions start a cycle at a high point. Sine functions start a cycle on the sinusoidal axis, heading up. ) The cycle will end one period later at PD. 5) Halfway between two high points will be a low point. Halfway between each high and low point, the graph will cross the sinusoidal axis. 6) After graphing the five critical points, sketch the graph through these five critical points. Example : Find the period, amplitude, frequency, phase displacement, and vertical displacement. Then use this information to find critical points and sketch the graph. Case : an equation y 5 3cos( 0 ) y VD A cos ( PD ) A 3,, VD 5, and PD 0 Amplitude = A 3 3 P 80 Frequency P cycle per degree 80 P step 80 5 Phase displacement: Vertical displacement: PD 0 shift right 0 VD 5 shift up 5 units, so the sinusoidal axis is at y 5 ChSec 9/6/07

2 Section. General Graphs (continued) Example continued: You want to find values of that make the argument equal to 0, 90, 80, 70, and. Remember, step 5, so the θ-values are 5 apart. 0 ) 0 ( 0 ) 90 ( 0 ) 80 ( 0 ) 70 ( 0 ) ( ( 0 ) y 5 3cos ( 0 ) y 5 3cos ( 0 ) Amplitude A 3 Axis y 5 Phase Displacement (PD) 0 right Vertical Displacement (VD) up 5 units Example : Find the period, amplitude, frequency, phase displacement, and vertical displacement. Then use this information to find critical points and sketch the graph. Case : an equation y 3 sin ( ) y VD A sin ( PD ) A,, VD 3, and PD Amplitude = A P ( ) 70 Frequency P cycle per degree 70 P step Phase displacement: Vertical displacement: PD shift left VD 3 shift down 3 units, so the sinusoidal axis is at y 3 ChSec 9/6/07

3 Section. General Graphs (continued) Example continued: You want to find values of that make the argument equal to 0, 90, 80, 70, and. ( 0 ) 0 ( ) ( ) 80 3 ( ) ( ) Remember, step 80, so the θ-values are 80 apart. ( ) y 3 sin ( ) y 3 sin ( ) Axis y ChSec 3 9/6/07

4 Section. General Graphs (continued) Example 3: For the sinusoid sketched, determine the period, frequency, amplitude, phase displacement, and vertical displacement. Then write an equation for the sinusoid. Case : a graph y Assume the graph is of a cosine function, since a cosine graph starts a cycle at a high point. y VD A cos ( PD ) One complete cycle begins at 5 and ends at. So, the period is The frequency is Frequency cycle per degree 60 The sinusoidal axis is halfway between the upper bound (U),, and the lower bound (L), 8. So, the vertical displacement is the average of and 8. U L VD 8 7 units The sinusoidal axis is y 7. The amplitude is the distance between the sinusoidal axis and the upper bound and it is a non-reflected cosine graph (thus, A is positive), A 7 35 units Assuming the graph to be of a cosine function, the phase displacement is 5, so PD 5. Since the period is 60, 60 6 So, an equation of the sinusoid graphed is y VD A cos ( PD ) y 7 35cos 6( 5 ChSec 9/6/07 )

5 Section. General Graphs (continued) Example : Draw a graph and find an equation of the sinusoid described. Case 3: the Amplitude,, PD, and VD 90, amplitude units, phase displacement (for a sine function) equals, vertical displacement 3 units. Assuming a non-reflected sine function, so A. 90 So, the equation is y VD A sin ( PD ) y 3 sin ) PD VD 3 You want to find values of that make the argument equal to 0, 90, 80, 70, and. ) 0 0 ) ) ) ) 90 0 P step 90.5 ) y 3 sin ) y 3 sin ) 3 Axis y All material has been taken from Trigonometry, by P. Foerster, 3 rd Edition ChSec 5 9/6/07

The Sine Function. Precalculus: Graphs of Sine and Cosine

The Sine Function. Precalculus: Graphs of Sine and Cosine Concepts: Graphs of Sine, Cosine, Sinusoids, Terminology (amplitude, period, phase shift, frequency). The Sine Function Domain: x R Range: y [ 1, 1] Continuity: continuous for all x Increasing-decreasing