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

1 Answer
Dec 16, 2015

Answer:

#((-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.