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

• MHB
• Elissa89
In summary, based on the graph of the function $y=-1/2[cos(x+pi)+cos(x-pi)]$, one could make a conjecture that $y=-cos(x)$. This is supported by the basic identity that shows the cosine curve reflected over the x-axis. Further analysis by David proved this conjecture using basic identities.
Elissa89
I don't even know what a conjecture is

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

Elissa89 said:
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 $y=cos(x)$

Last edited:
Elissa89 said:
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.

## 1. What does the graph of y=-1/2[cos(x+pi)+cos(x-pi)] look like?

The graph of this function is a sinusoidal curve with an amplitude of 1/2 and a period of pi. It is a combination of two cosine waves shifted by pi units in opposite directions.

## 2. What is the equation for the amplitude of this graph?

The amplitude of this graph is 1/2, which can be found by taking the absolute value of the coefficient in front of the cosine function.

## 3. What is the period of this function?

The period of this function is pi, which can be found by dividing 2pi (the period of a regular cosine function) by the coefficient in front of the x variable.

## 4. What are the x-intercepts of this graph?

The x-intercepts of this graph occur when the cosine functions are equal to 0. This happens at x = -pi/2 and x = 3pi/2.

## 5. What is the range of this function?

The range of this function is [-1/2, 1/2]. This is because the cosine function has a range of [-1, 1], and when multiplied by 1/2, it becomes [-1/2, 1/2].

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