Question

Write an arithmetic sequence that gives the nth positive x-intercept of the graph of f(x)=cos1/2x. Leave your answer in terms of π. WILL GIVE BRAINLIEST!

Answer 1

Let us consider the function given. It is $y=cos \frac{1}{2}x$. The coefficient of x (here 1/2) gives the frequency of the curve. What this means is that we see 1/2 a cycle between x=0 and x=$2 \pi$. Put another way, to see a full cycle we would need $\frac{2 \pi }{ \frac{1}{2} }=4 \pi$. The period of the function given is $4 \pi$.

Let us think about a cycle of cosine. It starts (x=0) and ends ($x=4 \pi$) its cycle at its highest value (here f(x)=1). It is at it's lowest (here f(x)=-1) in the middle ($x=2 \pi$) it is between the highest and the lowest that it crosses the x-axis.

That is, it has an x-intercept between 0 and $2 \pi$ That is, at $\pi$

The next one comes between $2 \pi$ and $4 \pi$. That is, at $3 \pi$

The attached shows the function and it's x-intercepts. You can see that they occur at: $\pi ,3 \pi ,5 \pi ,7 \pi ,9 \pi ,...$. This is the arithmetic sequence that contains the x-intercepts of the function.

As you can see the nth zero will occur at $n \pi$