Question

I was wondering if somebody would be so kind as to help understand this material better. Graph the following the equation:

y=-6 csc (1/3x)

I understand that the graph will be reflected about the x-axis and compressed by a factor of 6. What I don't understand is how the graph will be stretched by a factor of 1/3. Any help would be most appreciated, thank you.

Answer 1

Here are 2 plots
http://www.wolframalpha.com/input/?i=+++y%3D-6+csc+%28x%2F3%29%2C+y%3D+-6+csc%28x%29

for an idea on why it stretches:
say y=f(x) and we evaluate at x= 0, to get f(0)
and at x=1 we get f(1)

for the function y= f(x/3) , we get f(0) at x=0 and f(1/3) at x=1
we don't get f(1) until x=3 f(3/3) = f(1)
so with this function we had to move out to x=3 to match the first function. It is "stretched"

and of course y= f(3x) compresses or squeezes the function by a factor of 3 i.e. 3 copies in the space of 1 in the original f(x)

Answer 2

Thank you very much for your response, it has certainly helped me to better understand the material. At the end of the day, I think one of biggest issues is getting comfortable with radian measure. Any advice? Thanks again for your reply and have a great day!

Answer 3

radians can seem mysterious, but they are just a measure of an angle. If you were told a radian is 1/6 of the way around a circle you would not think them so complicated, but that is what they are (roughly).

What makes radians a bit of a nuisance is that there is not a nice rational number (like 6) radians in a circle, but 2pi radians. Pi is irrational (3.15159.... no pattern to the unending decimals). So rather than a nice number like 6 radians to a circle, we have 6.283185... radians (or more accurately) 2*pi radians
\[ 180º = \pi \text{ radians} \]
Here is a short discussion on why people use radians
http://mathforum.org/library/drmath/view/54181.html