1. Be able to write down the period and amplitude from a graph 2. Be able to state the period and amplitude from an equation 3. Be able to write down equation from the trig graph Sketch trig graphs

trig graphs The graph of a function which has a repeated pattern is called PERIODIC The graph of y = sinx y=cosx and y=tanx repeat themselves and are therefore periodic The period is the number of degrees or radians in each pattern The AMPLITUDE of the periodic graph is half the difference between the max and min values of the function ( the y coordinates)

Finding the period and the amplitude from graphs

y=sinx Period =360 amplitude =1 Features of the graph y =sinx Period =360 Amplitude=1 Max value = 1 when x = 90 Min value = when x=270 Graph cuts the x axis when x =0 180 and ( 90,1 ) (270,-1) Annotate graphs put important points on the graph

Features of the cosine graph y =cosx Period =360 Amplitude=1 Max value = 1 when x = 0 Min value = when x=180 Graph cuts the x axis when x =90 and270 and Annotate graph Amplitude=1 period (360,1) (0,1) (90,0) (180,-1) (270,0)

Features of the tangent graph y =tanx Period =180 Amplitude cannot be measured Max of min values cannot be measured Asymptotes at x =90 and x = 270 Graph cuts the x axis when x =0,180 and 360 and Annotate graph Period =180 (0,0) (180,0) (360,0) asymptotes

Graphs of the type y = 3sin x y = 3sinx Period = 360 Amplitude = 3 Max value = 3 when x = 90 Min value = -3 when x=270 Graph cuts the x axis when x =0 180 and 360 (90,3) (270,-3) (180,0) (360,0) (0,0) Amplitude =3 Period =360

Graphs of the type y=sin2x y =sin2x Period = 360/2 =180 Amplitude = 1 Max value = 1 when x = is 90/2 = 45 and Min value = when x=270/2 =135 and Graph cuts the x axis when x =0 180/2 and 360/2 0,90 and 180, 270, 360 (45,1) (90,0) (180,0)(270,0) (225,1) (135,-1) amplitude Period =180

Graphs of the type y=3sin2x y = 3sin2x Period = 360/2 =180 Amplitude = 3 Max value = 3 when x = is 90/2 = 45 and Min value = -3 when x=270/2 =135 and Graph cuts the x axis when x =0 180/2 and 360/2 0,90 and 180, 270, 360 Period =180 Amplitude =3 (45,3)(225,3) (135,-3) (315,-3) (90,0)(180,0) (270,0)

The general case y = a sinbx and y = acos bx Period = 360/b Amplitude = a Max value = a when x = is 90/b Min value = -a when x=270/b Graph cuts the x axis when x =0 180/b and 360/b The values are repeated every 360 /b degrees Period = 360/b Amplitude = a Max value = a when x = is 0 and 360/b Min value = -a when x=180/b Graph cuts the x axis when x =90/b and 270/b The values are repeated every 360 /b degrees Ex 4A p53 heineman

Graphs of the type y =asinbx +c y=acosbx + c Amplitude = a number of wavelengths in 360º = b shift along the y axis =c Period = 360/b y = asin(x+d) y=acos(x+d) Amplitude = a no. of wavelengths in 360º = 1 shift along the x axis = -d Period = 360

y =asinbx +c and y = acosbx =c This is like the graph of y = 3sin 2x but there is a shift of 1 up the y axis Period = 360/2 =180 Amplitude = 3 = (max – min ) /2 Max value = 4 when x = is 90/2 = 45 and Min value = -2 when x=270/2 =135 and to find where the graph cuts the x axis requires trig equations which are yet to be delt with period amplitude

Y = 3 sin (x + 30 ) Y = 3sin (x +30) is 30 to the left of y = 3 sin x a horizontal shift of 30 to the left a horizontal shift of 30 to the left

Period = 360 Amplitude = 3 Max value = 3 when x = = 60 Min value = -3 when x=270 –30 =240 Graph cuts the x axis when x =0-30, and Y=3sin(x + 30)

Y = 3 sin (x + 30 ) +1 Period =360 amplitude Period = 360 Amplitude = 3 = (5 - 2)/2 = ( max –min)/2 Max value = 4 when x = = 60 Min value = -2when x =270 –30 = 240 Graph cuts the x axis will be found from the trig equation 3 sin (x + 30 ) +1 = 0 (later work)

Eg2 for the function y= 3cos 2x + 1 state the period and the amplitude Period = 360 /2=180 Amplitude = 3

Given the trig graph find the equation y = sinbxy = a cos x + y = acos (x - d ) Amplitude a No. of wavelengths in 360 b Lift along the y axis c Shift along the x axis d abc d

The graph has an equation in the form y = asin bx or y = acos bx. Write an equation for the graph 2 wavelengths b = 2 Amplitude = 3 a = 3 Equation y = 3 cos 2x Graph like cos

Finding the equation of a trig graph 1 Number of wavelengths in 360 is Amplitude =(max- min)/2 (5-1)/2 = 4/2 =2 Eg2 The graph has an equation in the form y = asin bx +c or y = acos bx +c. Write an equation for the graph 4 b = 4 a = 2 Equation is y = 2sin 4x + 3 Graph is like a sin graph Shift =c = max -a = 5-2 = 3

Ex 4B p54 heineman Ex 3 p50 MIA

Sketching the graphs of trig functions Either use period amplitude and intersects with x and y axes Or use knowledge of transforming graphs learned in unit 1

Sketch and annotate the graph of y = 3 cos 2x-1 Amplitude =3Period = 360/2 = 180 Max value 3 –1 =2 when x = 0 and 360/2 = 180and Min values –3 –1 = -4when x = 180/2 = 90 an Add on the period, 180, to the first angles x xx xx (90,-4) (180,2) (270,-4)

Sketch and annotate the graph of y = 3 cos 2x-1 using composition of functions Start with y=cosx Draw y=cos2x compress (horizontally by a factor2) Draw 3cos2x ( stretch vertically by a factor3) Draw y=3cos2x –1 by sliding vertically down by 1 ex4B P54 HEINEMAN EX 3 P 50 MIA

MIA p 51 ex 3 q5, 6