tangent properties

[delish_fish]

one of my homework questions is to find the: domain, range, period, vertical asymptotes, zeros, symmetry and y-intercept
of this equation: y = -2tan(3x + 180°) + 3.
i've found the range which is y|yER and i think i've found the period... which is 180/3=60....
but the rest zeros, symmetry, y-intercept, and vertical asymptotes i'm lost on.
when i graph it on the graphing calculator, i just don't understand what lines i'm supposed to be using (because there's multiple lines when graphed if that makes any sense...)

i apologize if this is an elementary question, but it's really confusing to me. :\\

 

[galactus]

The vertical asymptotes occur wherever the function is undefined. Where the slope is infinite, so to speak.

\(\displaystyle \frac{d}{dx}[-2tan(3x+\pi)+3]=\frac{-6}{cos^{2}(3x)}\)

If you graphed this you can see how many VA's there will be. That is, they are close together.

\(\displaystyle cos^{2}(3x)=0\)

\(\displaystyle x=\frac{C\pi}{3}-\frac{\pi}{6}\) are the values of x that give an undefined result in said function.

To find successive vertical asymptotes, solve the inequality \(\displaystyle \frac{-\pi}{2}\leq 3x+\pi\leq \frac{\pi}{2}\)

This leads to \(\displaystyle \frac{-\pi}{2}\leq x\leq \frac{-\pi}{6}\).

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The zeros occur where the function crosses the x-axis.

\(\displaystyle -2tan(3x+\pi)+3=0\)

\(\displaystyle tan(3x+\pi)=\frac{3}{2}\)

\(\displaystyle 3x+\pi=tan^{-1}(\frac{3}{2})\)

\(\displaystyle 3x+\pi=\frac{\pi}{2}-tan^{-1}(\frac{2}{3})\)

\(\displaystyle x=\frac{C\pi}{3}+\frac{\pi}{6}-\frac{tan^{-1}(\frac{2}{3})}{3}\)

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The y intercept can be found by just setting x=0 in the function.

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The period of \(\displaystyle y=a\cdot tan(bx+c)\) is \(\displaystyle \frac{\pi}{|b|}\)

In this case, \(\displaystyle \frac{\pi}{3}\). You are correct if dealing with degrees.............but use radians.

By the way, was this problem given to you with 180 degrees?.

Degrees is an arbitrary measurement. In math, we use radians.

This is why I use Pi instead of 180.

 

[delish_fish]

yes, it was given to me in degrees. the lessons and examples were also in degrees. i've never learned how to do this with 'pi'.....
is it possible to do this in degrees though????

 

[tkhunny]

Time to learn...really! pi = 180º - Simple as that. This also allows you to get off the unit circle and wander through the rest of the world.