Graphing Rational Functions - Cool math Algebra Help Lessons

$Cool math fractal$

Your browser does not support the IFRAME tag.

$cool math anxiety survival and success guide$ $cool geometry art - fractals, polyhedra, more$ $other math stuff$
$cool math lessons$ $cool math games and practice problems$ $cool math games$ $free online math dictionary$
$Cool Math 4 Kids$

Coolmath Algebra
Graphing Rational Functions Lesson 5 - Putting It All Together
(page 1 of 3)
---- This algebra lesson shows how to put all the pieces together (x-intercept, y-intercept, vertical asymptotes, horizontal or slant asymptotes) and explains how to draw the graph for a rational function.

Your browser does not support the IFRAME tag.

	Your browser does not support the IFRAME tag.


Graphing Calculator	Scientific Calculator	<< a new window will open for these

OK... what were the 2 things?

	intercepts
	asymptotes

What were the 2 sentences?

	These guys are functions.
	Graphs hug asymptotes.

Let's do some graphing, baby!

Graph

2 things
and
2 sentences!

The y-intercept: Find f(0)

f ( 0 ) = ( 2 ( 0 )^2 + 5 ) / ( ( 0 )^2 - 25 ) = -1/5

( 0 , -1/5 ) ... nowhere else!

The x-intercept: Set numerator = 0 and solve

2x^2 + 5 = 0
Since we only deal with real numbers when graphing, this has no solution!
*Do you see why?

It will never cross the x-axis.
This is going to be very useful info!

Vertical asymptotes: Set denominator = 0 and solve

x^2 - 25 = 0 gives ( x - 5 ) ( x + 5 ) = 0 which gives x = 5 and x = -5

The lines x = -5 and x = 5

Horizontal asymptote:

Look at

( 2x^2 ) / ( x^2 ) = 2

The line y = 2

Now we're ready to graph... Remember that plotting points is for sissies, so use your brain!

First, let's get our intercepts and asymptotes on the graph:

intercept: ( 0 , -1/5 ) , no x intercept ... and asymptotes x = -5 , x = 5 , and y = 2

*Remember the the graph CANNOT cross the y-axis anywhere else and it CANNOT cross the x-axis at all!

Now, we use our brains and our sentences...

These are the three neighborhoods that this graph lives in:

The three neighborhoods in our grpah: left, middle, and right

Let's look at the left neighborhood:
*Remember your 2 sentences!

possible graphs for the left section ... since graphs hug asymptotes, our graph would need to do this or this

Since he's a function (and must pass the vertical line test), he can't do both.

What did we say about crossing the x-axis for this guy? (Look back if you need to.)

Yep -- he's not allowed to cross the x-axis! But, look at that bottom guy:

a possible graph for the left section ... If he has to hug the y = 2 asymptote and the x = -5 asymptote, he's got to cross the x-axis!

But, he's not allowed to -- so, he's got to live upstairs!

We can use the same reasoning for the right neighborhood:

the left and right sections of the graph of f ( x ) = ( 2x^2 + 5 ) / ( x^2 - 25 )

What about the middle neighborhood?

He's got to hug these walls. (the asymptotes x = -5 and x = 5)

And remember, he's not allowed to cross the x-axis! So, he CAN'T do any of these things:

possibilities for the middle section of the graph

What's left?

graph of f ( x ) = ( 2x^2+ 5 ) / ( x^2 - 25 ) ... DONE!

YOUR TURN:

Graph

f ( x ) = ( x^2 - x - 6) / ( x^2 - 1 )

(You already have all the pieces!)

Hey, you can check it on the graphing calculator! Enter it in like this:

Continued on the next page

The printing and distribution and/or downloading of these lessons is strictly prohibited.


Graphing Calculator	Scientific Calculator	<< a new window will open for these

||||| copyrights and fair use of these lessons |||||

.....::::::::::::::: HELP SUPPORT COOLMATH :::::::::::::::.....
::::::: link to us ::::::: advertise with us ::::::: why we have ads ::::::: cool math games store :::::::

|| about us || about our Cool Math Games || safe surfing || privacy policy || terms of use || copyrights & fair use ||

|| cool math pre-algebra || cool math algebra || cool math precalculus / calculus || cool math anxiety survival || geometry / art / fractals / polyhedra ||
|| free online brain bender and jigsaw puzzles || cool math books || other math stuff ||

|| cool math lessons: how to... || cool math games & practice problems || cool math 4 kids lessons and games || cool math games for arithmetic ||
|| free online math dictionary || geometry / trigonometry reference || cool math for teachers || cool math for parents || cool math games ||

|| Coolified Games || Cool Math 4 kids || Spike's Game Zone || Finance Freak || Totally Stressed Out || Science Monster lessons ||