4.5 and 4.6 Graphs of Trig Functions Obj: to identify amplitude and period of trig functions and translate them

On your graphing calc…. Graph y=sin(x)

General form of Sine function |A| is the amplitude (how high up) Period of the function is T=2  /B Phase shift (starting point) of the function is C/B – Positive phase shift means it’s moved to the left – Negative phase shift means it’s moved to the right. D is the vertical shift up or down. y = Asin(Bx - C) +D

Five steps to graph 1.Identify amplitude and period. 2.Find the x values for the five key points (divide period by 4) 3.Find the values of y for the five key poitns. 4.Connect the 5 key points with a smooth curve and graph one complete cycle. 5.Extend graph to the left and right.

Example Graph Y = 4 sin (3x) 1.Identify amplitude and period. 2.Find the x values for the five key points (divide period by 4) 3.Find the values of y for the five key points. 4.Connect the 5 key points with a smooth curve and graph one complete cycle. 5.Extend graph to the left and right.

Example Graph Y = 2 sin (2x-2  /3)+1 1.Identify amplitude, period, phase shift. 2.Find the x values for the five key points (divide period by 4) 3.Find the values of y for the five key points. 4.Connect the 5 key points with a smooth curve and graph one complete cycle. 5.Extend graph to the left and right.

General form of the cosine function y = Acos(Bx - C) + D Cosine graphs look the same as sine graphs but the basic function is p.s. to the right 90 deg. P.S., period, amplitude are found the same way.

Try one… Determine the amp, period, and p.s. of

One more Graph one period of the function

4.6 Graphs of other trig functions Graphing y = A tan (Bx – C)

Tangent functions Have asymptotes because they’re sometimes undefined 1.Find 2 consec asymp by finding an interval containing one period. 2.Identify an x intercept, midway between the asymptotes 3.Find the points on the graph ½ and ¾ of the way b/t consec. Asymp. These have y-coord of –A and A, respectively. 4.Use steps 1-3 to graph one full period.

Example – Graph y = 2 tan x/2 A=2,B=1/2, C=0 1.Find 2 consec asymptotes: 2.X intercept is at x= 0 midway b/t consec asymp. 3.Find pts. ¼ of the way and ¾ of the way between the asymptotes (-  /2 and  /2) Coord are: (-  /2,-2) and (  /2,2) 4.Graph full period.

Example – Graph y = 3 tan x/4 A=,B=, C= 1.Find 2 consec asymptotes: 2.X intercept is midway b/t consec asymp. 3.Find pts. ¼ of the way and ¾ of the way between the asymptotes Coord are: 4.Graph full period.

Graph

Graphing cotangent Y=Acot(Bx-C) 1.Find 2 consec asymp by finding an interval containing one period. 2.Identify an x intercept, midway between the asymptotes 3.Find the points on the graph ½ and ¾ of the way b/t consec. Asymp. These have y-coord of –A and A, respectively. 4.Use steps 1-3 to graph one full period.

Graph Y=Acot(Bx-C) 1.Find 2 consec asymp by finding an interval containing one period. 2.Identify an x intercept, midway between the asymptotes 3.Find the points on the graph ½ and ¾ of the way b/t consec. asymp. These have y-coord of –A and A, respectively. 4.Use steps 1-3 to graph one full period.

Using sin and cos to graph csc and sec