warmup alg 2 6 mar 2012

Warmup Alg 2 6 Mar 2012

Warmup Alg 2 7 & 8 Mar 2012

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• Chapter 8: Rational Functions• Can you graph simple rational functions?

Section 8.2: Simple Rational Functions and

their graphs

Vocabulary

Domain

Range

The places along the x axis where the graph lives

The places along the y axis where the graph lives

Vocabulary

Rational Expression

Asymptote

Vertical

Horizontal

A quotient of 2 polynomials

An invisible line that the graph gets close to, but never touches.

Vocabulary

Vertical Asymptote Occurs anywhere the fraction is divided by zero.

You must, MUST, MUST check to find out where the denominator equals zero.

This is NOT optional.

Checking for Vertical Asymptotes

xx662 3 Is this a Rational function?

Yes, the numerator is cubic, the denominator is linear

Where does the denominator equal zero?6x Take the bottom= 0

Set it equal to zero and solve.

x = 0

There is a vertical asymptote at x = 0

Checking for Vertical Asymptotes

732285

2

23

xxxx x

Is this a Rational function?

No, the 2x messes it up!

Finding the domain

12662 3

xx What is the domain of the

expression?

Where does the denominator equal zero?

6x -12 Take the bottom= 0Set it equal to zero and solve.

6x = 12

There is a vertical asymptote at x = 2 x = 2

The domain is: All Reals except x = 2

Graphing Simple Rational Functions

xy 1 The parent function for all

rationals.

Where does the denominator equal zero?

x Take the bottom= 0Set it equal to zero and solve.

There is a vertical asymptote at x = 0

The domain is: All Reals except x = 0

xy 1

x= 0 is the vertical asymptote, and y=0 is the horizontal asymptote

The domain is all reals and x ≠ 0

The range is all reals and y ≠ 0

Textbook notes

xy 6

x= 0 is the vertical asymptote and y =0 is the horizontal asymptote.

The domain is all reals except x = 0

The range is all reals except y = 0

xy 12

x= 0 is the vertical asymptote and y = 0 is the horizontal asymptote.

The domain is all reals except x = 0

The range is all reals except y = 0

xy 4

x= 0 is the vertical asymptote and y = 0 is the horizontal asymptote.

The domain is all reals except x = 0

The range is all reals except x = 0

Graphing Simple Rational Functions

xay What does the “a” do to the

equation?

How did the graphs change as “a” went from 1 to 12?

Just like all the other translations, “a” makes it skinny or fat, and a negative “flipped it” around the X AXIS.

Graphing Simple Rational Functions

khxay

What do you think the “h” and the “k” do to the graph?

If you said, “The same thing as with radicals and quadratics and polynomials,” you are RIGHT!

“h” moves it left and right, (but opposite to the sign” and “k” moves it up and down!

Textbook notes

32

x

y

x= -3 is the vertical asymptote and y = 0 is the horizontal asymptote.

The domain is all reals except x = -3

The range is all reals except x = 0

432

x

y

x= -3 is the vertical asymptote and y = 4 is the horizontal asymptote.

The domain is all reals except x = -3

The range is all reals except x = 4

Graphing Rational Functions

dcxbaxy

Now they are a little more complex.

The numerator and denominator are both lines, that is very important.

The vertical asymptote is at x = -d/c

The horizontal asymptote is at y = a/c

432

xxy

x = 4 is the vertical asymptotey=2 is the horizontal asymptote.

The domain is all reals except x = 4

The range is all reals except y = 2/1 = 2

Textbook notes

Assignment

Section 8.2: 11 – 18, 29 – 34