Monday, April 9, 2018

Week of April 9, 2018

Algebra 1

Complete your spring make -up packet and unit 4 study guide.

Algebra 2

Graphing Rational Functions 


Graph
 		f ( x ) = ( 2x^2 + 5 ) / ( x^2 - 25 )
 
2 things
and
2 sentences!
 

1
 

The y-intercept: Find f(0)
 
f ( 0 ) = ( 2 ( 0 )^2 + 5 ) / ( ( 0 )^2 - 25 ) = -1/5
( 0 , -1/5 ) ... nowhere else!

2
 

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!

3
 

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

4
 
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!Source:http://www.coolmath.com/algebra/23-graphing-rational-functions/05-graphing-rational-functions-all-parts-01
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