Sketch without using technology?

Question

0.16t/(t^2+t+2)

How do u get that weird looking curve without technology?

Answer 1

To get a rough sketch of a rational function, you need to find a few things:

-Vertical Asymptotes
-End behavior/Horizontal Asymptotes

-Zeroes

 

To find the vertical asymptotes of a rational function, determine what values make the denominator zero. Here, where does:

t² + t + 2 = 0

This quadratic does not factor, and if you use the quadratic formula, you'll get two imaginary answers. The denominator is always positive, never zero, and thus the function has no vertical asymptotes.

 

To determine the end behavior and possible horizontal asymptotes of a rational function, consider the ratio of the leading terms of the numerator and denominator. In this case, you have

0.16t/t²

Because the degree of the denominator is higher, the horizontal asymptote is y = 0. This means that as t increases toward +∞ and -∞, the function approaches 0.

 

To find the zeroes of the function, set the rational function equal to zero. If your numerator is not zero, you cannot divide by anything to make it zero, so you only need to consider the numerator.

0.16t = 0

t = 0 is your only zero.

 

At this point, you just need to plug in some points to get an idea of what the values of the function will be on either side of your zeroes. Since you have a single zero, you should choose a couple points on each side of 0, and find the values to plot. Then you can connect them and sketch, remembering to take the end behavior into account. t = -2, -1, 1, 2 would be decent choices to pick.

 

I'm assuming you have not learned derivatives yet. If you have, then you can use those methods to find more specific interesting points and curvature of the graph.

 

I hope this help, if you have further questions, please comment.