MHB Graph the function y=-1/2[cos(x+pi)+cos(x-pi)] and make a conjecture

[Elissa89]

I don't even know what a conjecture is

y=-1/2[cos(x+pi)+cos(x-pi)]

 

[DavidCampen]

I don't even know what a conjecture is

y=-1/2[cos(x+pi)+cos(x-pi)]

A conjecture is something that you think might be true.

We know that $cos(x+\pi)$ = $cos(x-\pi)$
so $y=-\frac{1}{2}[2cos(x+\pi)] = -cos(x+\pi)$
also, since we know that $cos(x+\pi) = -cos(x)$ we would have $y=cos(x)$

so maybe that is supposed to be the conjecture, that $y=cos(x)$

 

[skeeter]

I don't even know what a conjecture is

y=-1/2[cos(x+pi)+cos(x-pi)]

https://en.wikipedia.org/wiki/Conjecture

The directions in the title of your post say to graph the function and make a conjecture based on what you see in the graph.

Note the graph of $y$ shows the basic cosine curve reflected over the x-axis ... in other words, one could make a conjecture that $y=-\cos{x}$.

David went a step further and proved the conjecture using basic identities.

[DESMOS]advanced: {"version":7,"graph":{"xAxisStep":1.5707963267948966,"yAxisStep":1,"squareAxes":false,"viewport":{"xmin":-6.564569536423841,"ymin":-2.039999999999999,"xmax":6.435430463576159,"ymax":1.9600000000000009}},"expressions":{"list":[{"type":"expression","id":"graph1","color":"#2d70b3","latex":"y=\\frac{1}{2}\\left(\\cos\\left(x+\\pi\\right)+\\cos\\left(x-\\pi\\right)\\right)"},{"type":"expression","id":"2","color":"#388c46"}]}}[/DESMOS]