GRAPHING RATIONAL FUNCTIONSADV122

Warm Up

Graph the function

GRAPHING RATIONAL FUNCTIONSADV122

We have graphed several functions, now we are adding one more to the list!

Graphing Rational Functions

GRAPHING RATIONAL FUNCTIONSADV122

Parent Function:

GRAPHING RATIONAL FUNCTIONSADV122

f(x) = + ka

x – h

(-a indicates a reflection in the x-axis)

vertical translation(-k = down, +k = up)

horizontal translation(+h = left, -h = right)

Pay attention to the transformation clues!

Watch the negative sign!! If h = -2 it will appear as x + 2.

GRAPHING RATIONAL FUNCTIONSADV122

Asymptotes

Places on the graph the function will approach, but will never touch.

GRAPHING RATIONAL FUNCTIONSADV122

f(x) =

1x

Vertical Asymptote: x = 0Horizontal Asymptote: y = 0

Graph:

A HYPERBOLA!!

No horizontal shift.No vertical shift.

GRAPHING RATIONAL FUNCTIONSADV122

W look like?

GRAPHING RATIONAL FUNCTIONSADV122

Graph: f(x) = 1

x + 4

Vertical Asymptote: x = -4

x + 4 indicates a shift 4 units left

Horizontal Asymptote: y = 0

No vertical shift

GRAPHING RATIONAL FUNCTIONSADV122

Graph: f(x) = – 31x + 4

x + 4 indicates a shift 4 units left

–3 indicates a shift 3 units down which becomes the new horizontal asymptote y = -3.

Vertical Asymptote: x = -4

Horizontal Asymptote: y = 0

GRAPHING RATIONAL FUNCTIONSADV122

Graph: f(x) = + 6x

x – 1

x – 1 indicates a shift 1 unit right

+6 indicates a shift 6 units up moving the horizontal asymptote to y = 6

Vertical Asymptote: x = 1

Horizontal Asymptote: y = 1

GRAPHING RATIONAL FUNCTIONSADV122

You try!!

2.

GRAPHING RATIONAL FUNCTIONSADV122

How do we find asymptotes based on an equation only?

GRAPHING RATIONAL FUNCTIONSADV122

Vertical Asymptotes (easy one) Set the denominator equal to zero

and solve for x. Example:

x-3=0 x=3

So: 3 is a vertical asymptote.

GRAPHING RATIONAL FUNCTIONSADV122

Horizontal Asymptotes (H.A) In order to have a horizontal

asymptote, the degree of the denominator must be the same, or greater than the degree in the numerator.

Examples: No H.A because Has a H.A because 3=3. Has a H.A because

GRAPHING RATIONAL FUNCTIONSADV122

3 cases

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If the degree of the denominator is GREATER

than the numerator.

The Asymptote is y=0 ( the x-axis)

GRAPHING RATIONAL FUNCTIONSADV122

If the degree of the denominator and

numerator are the same: Divide the leading coefficient of the numerator by the leading coefficient of the denominator in order to find the horizontal asymptote.

Example: Asymptote is 6/3 =2.

GRAPHING RATIONAL FUNCTIONSADV122

If there is a Vertical Shift The asymptote will be the same

number as the vertical shift. (think about why this is based on the

examples we did with graphs)

Example:

Vertical shift is 7, so H.A is at 7.

GRAPHING RATIONAL FUNCTIONSADV122

Homework

http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Graphing%20Simple%20Rational%20Functions.pdf