<p>Hey, I read the explanation for it on the sticky at the top of the page but I still don't understand it. Can anyone else make it clearer and help me out?</p>
<p>There are a few ways to attack this problem, all fairly similar in nature. I'll give two descriptions, one which focuses on the numbers alone and one which focuses on the graph alone; hopefully, one of the two will make sense to you.</p>
<p>Numerical method: Analyze the portion of the graph that they give you. The question is asking for locations where f(x) = 0 (or, equivalently, where the graph crosses the x axis), so just look at what is there for the moment and ignore the bit about the values from 5 to 12. A quick look at the graph shows four points at which the graph intercepts the x-axis. We can't really extract exact values for x at some of those intercepts, but it does not matter--just assign a value that is reasonably close and you'll be fine. These four values for x, then, are around 1.5, 2.5, 3, and 4.5. Writing this same information another way, f(1.5) = 0, f(2.5) = 0, f(3) = 0, and f(4.5) = 0. Now, we can apply this statement that the problem gives us: "f(x + 5) = f(x) for all values of x." Applying this little sentence to one of our values of x, 0 = f(1.5) = f(1.5 + 5) (or f(6.5)) = f(1.5 + 5 + 5) (or f(11.5)) = f(1.5 + 5 + 5 + 5) (or (f(16.5)) ...and so on. Thankfully, we are only interested in values of x between 0 and 12, so we can stop applying this statement as soon as the number in the function goes beyond 12. So, applying it to each of our points:
f(1.5) = f(6.5) = f(11.5) (then 16.5 is too big, so we stop)
f(2.5) = f(7.5) (12.5 is too big)
f(3) = f(8) (13 is too big)
f(4.5) = f(9.5) (14.5 is too big)
So, our solutions for x include 1.5, 6.5, 11.5, 2.5, 7.5, 3, 8, 4.5, and 9.5, for a total of 9 solutions.</p>
<p>Graphing method: The biggest hurdle in understanding this problem is getting a feel for what the phrase "f(x + 5) = f(x) for all values of x" means. If you stop to think about it for a moment, the description is telling you that the graph repeats itself over and over every 5 units on the graph. Using this information, then, you can draw the graph out to x=12 relatively quickly. What you'll get will be something like this: <a href="http://aycu33.webshots.com/image/36672/2005300529706718163_rs.jpg%5B/url%5D">http://aycu33.webshots.com/image/36672/2005300529706718163_rs.jpg</a>
With a graph drawn out like this, the problem simply becomes an issue of counting the number of times the graph crosses the x axis (circled in red on the image). Provided your graph is correct, you should come out with 9 solutions.</p>
<p>Let me know if you're still having trouble with it, and sorry if these descriptions don't help much.</p>
<p>Oh ok thanks man. The numerical helped a lot; I'm terrible with graphs but that picture helped a lot with the visualization.</p>