Graphing Trig Functions. Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions.

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Jody Webster
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1 Graphing Trig Functions Name: Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions. y = sinx (0,) x 0 sinx (,0) (0, ) (,0) /2 3/2 /2 3/2 2 x cosx 2 y = cosx 0 /2 /2 3/2 2 3/2 2 Apr 29 3:37 PM Properties of y = sinx and cosx -The domain of each function is. -The range of each function is. -The of each function is half the difference of the maximum and minimum. -Each function is, which means its graph has a repeating pattern. The shortest repeating portion of the graph is called the. The horizontal length of each cycle is called the. -The period of each function is. Apr 29 3:49 PM

2 Examples: Determine the amplitude and period of each function graphed below..) 5 /4 /2 3/4 5/4 3/2-5 2.) Apr 29 3:59 PM Amplitude and Period: The amplitude and period of the graphs y = asinbx and y = acosbx are as follows: Amplitude = a Period = 2 b Examples: Graph the following..) y = 4sinx 2.) y = cos4x Apr 29 3:53 PM 2

3 Examples: Graph the following..) y = 2sin¼x 2.) y = 2cosx Apr 29 4:0 PM Let's graph y = tanx by filling out the table below. (0,) (,0) (,0) (0, ) x 0 /4 /2 3/4 5/4 3/2 7/4 2 tanx /2 3/2 2 Apr 29 4:6 PM 3

4 Period and Vertical Asymptotes: The period and vertical asymptotes of the graph of y = atanbx are as follows: - The period is. b - The vertical asymptotes are at odd multiples of 2b Examples: Graph one period of the functions below..) y = 2tan3x 2.) y = 4tan2x Apr 29 4:9 PM Translations/Reflections of Trig Functions (0,) x 0 /2 -sinx (,0) (0, ) (,0) 3/2 y = -sinx /2 3/2 2 2 x -cosx 0 /2 3/2 2 y = -cosx /2 3/2 2 Apr 29 3:37 PM 4

5 Along with reflections, graphs of trig functions can also translate left/right and up/down. Translations of Sine and Cosine Graphs To graph y = asin b(x - h) + k or y = acos b(x - h) + k, follow these steps:.) Identify the amplitude a, the period 2/b, the horizontal shift h,the vertical shift k and note any reflection. 2.) Draw the horizontal line y = k, which is called the midline. 3.) Find the five key points by translating the key points of y = asinbx and y = acosbx in the following order: -horizontally h units -reflect (if necessary) 4.) Draw the graph through the five translated key points. Apr 29 5:27 PM Examples:.) Graph y = sin4x ) y = 4cos(x - ) Apr 29 5:42 PM 5

6 3.) y = sin2(x + /2) ) y = -2sin[(/2)(x - )] Apr 29 5:44 PM Examples:.) Write a cosine equation that represents the graph. -/4 /2-2.) Write a sine equation that represents the graph Apr 29 5:50 PM 6

7 Graphing Reciprocal Trig Functions y = cscx y = secx Mar 2:6 PM Examples Graph..) y = 2csc(x - ) 2.) y = -sec[2(x - /2)] + Mar 2:20 PM 7

8 y = cotx Examples Graph..) y = 2cotx + Mar 2:6 PM 2.) y = cot(x - /4) + 3.) y = -tan[2(x + /8)] - Mar 2:48 PM 8

9 Graph Trig Functions Homework Name: Graph the following trig functions. Label!.) y = 2sinx 2.) y = -cos2x 3.) Fill in the blank. The graphs of the functions y = sinx and y = cosx both have a of 2. They both have an of. Apr 29 6:2 PM 4.) Write both a sine and cosine equation of the graph below. 2 5.) Graph y = -4sinx. Label! Apr 29 6:23 PM 9

10 6.) Graph one period of y = 4tanx. Label! 7.) Graph one period of y = 3tan2x. Label! Fill in the blanks. 8.) The graph of y = cos2(x - 3) is the graph of y = cos2x translated units to the right. The graph of y = cos2x + is the graph of y = cos2x translated units up. Apr 29 6:28 PM 9.) Graph y = 3cos(x + 3/2) -. Label! 0.) Write a sine equation for the graph below. 4 8 Apr 29 6:33 PM 0

11 Graph..) y = -4cos(x + ) - 2.) y = 2sin[2(x - /2)] + Mar 2:20 PM Graph. 3.) y = 3sec(x + ) 4.) y = csc[4(x - /2)] + Mar 2:20 PM

12 Graph. 5.) y = cotx - 6.) y = 2tan[(x + /2)] Mar 2:20 PM 7.) Write a sine function with a period of, an amplitude of 3 and a vertical shift up 2. 8.) Write a cosine function with a period of /2, a reflection over the x-axis, an amplitude of 4 and a vertical shift down 2. 9.) Each branch of y = secx and y = cscx is a curve. Explain why these curves cannot be parabolas. Hint: Do parabolas have asymptotes? Mar :02 PM 2

Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions In this section, we will look at the graphs of the other four trigonometric functions. We will start by examining the tangent