How do you find the critical points to graph y=2cos(1/2x+pi/3)-1?

1 Answer
Dec 16, 2015

((-2pi)/3,1), (pi/3,-1), ((4pi)/3,-3), ((7pi)/3,-1), ((10pi)/3,1)

Explanation:

Start by factoring out 1/2 in the bracketed part of the equation:

y=2cos(1/2x+pi/3)-1
becomes:
y=2cos(1/2(x+(2pi)/3))-1

By the five-point method, you need five points to graph the function, y=2cos(1/2(x+(2pi)/3))-1. To find the five points, first find the five points for its parent function y=cosx. The five main points, 0, pi/2, pi, (2pi)/3, and 2pi, including their corresponding y values are listed below.

http://www.pinkmonkey.com/studyguides/subjects/trig/chap5/t0505401.asp

Now that you have the five points for the parent function, use the mapping rule to apply transformations in order to find the five points for the transformed function, y=2cos(1/2(x+(2pi)/3))-1.

Mapping rule: (color(red)2x-color(blue)((2pi)/3),color(orange)2y-color(green)1)

y=2cos(1/2(x+(2pi)/3))-1
Point 1. ((color(red)2)0-color(blue)((2pi)/3), color(white)(xxx)(color(orange)2)1-color(green)1) rArr color(white)(ixxxxxx)((-2pi)/3,1)
Point 2. ((color(red)2)pi/2-color(blue)((2pi)/3), color(white)(xx)(color(orange)2)0-color(green)1) rArr color(white)(xxxxxxx)(pi/3,-1)
Point 3. ((color(red)2)pi-color(blue)((2pi)/3), color(white)(xxx)(color(orange)2)(-1)-color(green)1) rArr color(white)(xxx)((4pi)/3,-3)
Point 4. ((color(red)2)(3pi)/2-color(blue)((2pi)/3), color(white)(ix)(color(orange)2)0-color(green)1) rArr color(white)(xxxxxxx)((7pi)/3,-1)
Point 5. ((color(red)2)2pi-color(blue)((2pi)/3), color(white)(xx)(color(orange)2)1-color(green)1) rArr color(white)(xxxxxxx)((10pi)/3,1)

https://www.desmos.com/screenshot/s5v18fxvyj

Zoom in to check the five main points shown on the graph.