Section+3.6

Section 3.6 (Friday, October 28th)

Graph media type="custom" key="11213836"

//a) Find where the function is zero// //b) Find where the function is undefined// //c) Find asymptotes and which direction they go// //c) Graph function with asymptotes and bounces//

a) To find roots, find when the **numerator** media type="custom" key="11538702" **is equal to zero.**

Using your calculator, you can find one root at media type="custom" key="11538704" Then, **use synthetic division** to find another.

__2|__ 3 -30 96 -96 __2|__ 0 6 -48 96 __2__| 3 -24 48 0 media type="custom" key="11148504"
 * Note**: Remainder must be 0, if it isn't you know you did something wrong

So, there is another zero at media type="custom" key="11538710" Therefore, the graph will ONLY cross the x axis at media type="custom" key="11538718" Furthermore, the zero at media type="custom" key="11538774" cancels and won't appear on the graph of the function because we will soon find that the function is undefined at media type="custom" key="11538780".
 * Note**: Since media type="custom" key="11538772" is a double root, the function will bounce at media type="custom" key="11538714"

b) To find asymptotes (where the function is undefined), find when the **denominator**, media type="custom" key="11148544" **is equal to zero.**

media type="custom" key="11148566"

Again, **use synthetic division**.

__2|__ 1 1 -16 20 __2|__ 0 2 -48 -20 __2__| 1 3 -10 0 media type="custom" key="11148556"

So the graph is **undefined** at media type="custom" key="11148740" and media type="custom" key="11148742" Since the graph is undefined at media type="custom" key="11539000", the zero at media type="custom" key="11539006"cancels out. In some cases, you will have values that are both zeros and undefined values. When this happens, the zero cancels out because if the graph is undefined at that value, it is impossible to have a zero there.

Zeros of function: media type="custom" key="11538782" Undefined x values: media type="custom" key="11538790"

c) Since the graph is undefined at media type="custom" key="11148750" and media type="custom" key="11148756", **two vertical asymptotes** form.

To find the direction of the asymptote at media type="custom" key="11148734", **plug media type="custom" key="11538798" into the equation** media type="custom" key="11538904" You will find that the value is a big, positive number, meaning that the function goes up along the **vertical asymptote** at media type="custom" key="11538732"

To find the direction of the asymptote at media type="custom" key="11148686", **plug media type="custom" key="11538802" into the same equation.** You will find that the value is also a big, positive number, so the function goes up along the **vertical asymptote** at media type="custom" key="11538744" Furthermore, we know that a **horizontal asymptote** forms at media type="custom" key="11148696" because when you plug in an extremely large number, the graph will be about media type="custom" key="11538746".

As for the space between the two asymptotes, or when media type="custom" key="11538858", we know that it can't cross the x intercept because the only root is at media type="custom" key="11538862". Therefore, a **parabola** must be formed and both ends must go down along the vertical asymptotes at media type="custom" key="11538762" and media type="custom" key="11538768".

d) Wolfram Alpha Graph


 * Important things to understand:**

//Remainder when doing synthetic division must be zero, if it isn't you know you did something wrong.// //A graph never crosses a vertical asymptote, it can only cross a horizontal asymptote when x is small.// //If there is a double root, the graph will bounce at that x value.// //To find the direction of asymptotes, plug in extremely small or large numbers.// //Be sure to check both sides of an asymptote to see what's happening.// //Don't worry about finding where the graph turns around, that is calculus.//