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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)