Section 9.2

Graphing Simple Rational Functions

Graphing a Simple Rational Function

A rational function is a function of the form where p (x) and q (x) are polynomials and q (x) 0.

p (x)q (x)f (x) =

For instance consider the following rational function: 1x

y =

The graph of this function is called a hyperbola and is shown below.

Graphing a Simple Rational Function

1413

12

12

– 4

–2

–1– 1

– 2

– 3

x y

–

–

–

–

4

2

1 1

2

3

x y1413

12

12

The x-axis is a horizontal asymptote.

The y-axis is a vertical asymptote.

The domain and range are all nonzero real numbers.

The graph has two symmetrical parts called branches. For each point (x, y) on one branch, there is a corresponding point (–x, –y) on the other branch.

Notice the following properties:

Graphing a Simple Rational Function

To draw the graph, plot a couple of points on each side of the vertical asymptote.

All rational functions of the form have graphs that are

hyperbolas with asymptotes at x = h and y = k.

ax – h y = + k

Then draw the two branches of the hyperbola that approach the asymptotes and pass through the plotted points.

Graphing a Rational Function

–2x + 3 Graph . State the domain and range.y = – 1

the asymptotes at x = –3 and y = –1.Draw

two points to the left of the vertical asymptote, such as (–4, 1) and (–5, 0), and two points to the right, such as (–1, –2),

and (0, ). 53

–

Use the asymptotes and plotted points to draw the branches of the hyperbola.

The domain is all real numbers except –3.

SOLUTION

Plot

The range is all real numbers except –1.

Graphing a Rational Function

All rational functions of the form have graphs that are hyperbolas.a x + bc x + d

y =

The vertical asymptote occurs at the x-value that makes the denominator zero.

The horizontal asymptote is the line . ac

y =

two points to the left of the vertical

asymptote, such as (0, ) and (1, –1), and two

points to the right, such as (3, 2), and (4, ). 54

14

–

Graphing a Rational Function

Graph . State the domain and range.x + 1

2x – 4 y =

SOLUTION

Draw

Use the asymptotes and plotted points to draw the branches of the hyperbola.

The domain is all real numbers except 2.

the asymptotes. Solve 2x – 4 = 0 for x to find the vertical asymptote x = 2.The horizontal

asymptote is the line .12

=ac

y =

The range is all real numbers except .12

Plot

Graphing a Rational Function

Find the vertical and horizontal asymptotes for the graph.

3x+76x-6 y =

SOLUTION

Draw the asymptotes. Solve 6x - 6 = 0 for x to find the vertical asymptote x = 1.The horizontal

asymptote is the line .12

=ac

y =

Using Rational Functions in Real Life

For a fundraising project, your math club is publishing a fractal art calendar. The cost of the digital images and the permission to use them is $850. In addition to these “one-time” charges, the unit cost of printing each calendar is $3.25.

Write a model that gives the average cost per calendar as a function of the number of calendars printed.

SOLUTIONThe average cost is the total cost of making the calendars divided by the number of calendars printed.

Verbal Model

…

Average cost

One-time charges + Unit cost • Number printed

Number printed=

Using Rational Functions in Real Life

For a fundraising project, your math club is publishing a fractal art calendar. The cost of the digital images and the permission to use them is $850. In addition to these “one-time” charges, the unit cost of printing each calendar is $3.25.

Labels Average cost = A

One-time charges = 850

Unit cost = 3.25

Number printed = x

(dollars)

(dollars per calendar)

(dollars per calendar)

(calendars)

…

Algebraic Model

A850 + 3.25 x

x=

Write a model that gives the average cost per calendar as a function of the number of calendars printed.

For a fundraising project, your math club is publishing a fractal art calendar. The cost of the digital images and the permission to use them is $850. In addition to these “one-time” charges, the unit cost of printing each calendar is $3.25.

Use the graph to estimate the number of calendars you need to print before the average cost drops to $5 a calendar.

The graph of the model is shown. The A-axis is the vertical asymptote and the line A = 3.25 is the horizontal asymptote.

The domain is x > 0 and the range is A > 3.25.

When A = 5 the value of x is about 500. So you need to print about 500 calendars before the average cost drops to $5 per calendar.

Using Rational Functions in Real Life

For a fundraising project, your math club is publishing a fractal art calendar. The cost of the digital images and the permission to use them is $850. In addition to these “one-time” charges, the unit cost of printing each calendar is $3.25.

Describe what happens to the average cost as the number of calendars printed increases.

As the number of calendars printed increases, the average cost per calendar gets closer and closer to $3.25.

For instance,when x = 5000 the average cost is $3.42, and when x = 10,000 the average cost is $3.34.

Using Rational Functions in Real Life

Assignment

Section 9.2: page 543 – 544 # 12 – 18 (every 3rd), 20 –

22, 24 – 39 (every 3rd)