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
Examples: Determine the amplitude and period of each function graphed below..) 5 /4 /2 3/4 5/4 3/2-5 2.) 2 4 - 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
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
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
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 + 3 2.) y = 4cos(x - ) Apr 29 5:42 PM 5
3.) y = sin2(x + /2) - 3 4.) 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. 2-4 4 Apr 29 5:50 PM 6
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
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
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
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
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
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