Chapter 9: Rational Equations and Functions
Chapter 9.1: Inverse and Joint Variation
Direct and Inverse Variation
Example: Do x and y show direct variation, inverse variation, or neither?
Example: and vary inversely, and
when
Write an equation that relates and .
Find y when
.
Example: and vary inversely, and
Find an equation that relates and .
Find y when
.
when
.
.
Example: The volume of gas in a container varies inversely with the amount of pressure. A gas has
⁄ . Write a model relating volume and pressure.
volume
at a pressure of
Example: Do these data show inverse variation? Is so, find the model.
Joint Variation
Example: Write an equation.
varies directly with and inversely with
varies inversely with
varies directly with
varies jointly with
varies inversely with
.
.
and inversely with .
and .
and .
Example: The ideal gas law states that the volume V (in liters) varies directly with the number of
molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in
kilopascals). The constant of variation is denoted by R and is called the universal gas constant.
Write an equation for the ideal gas law.
Estimate the universal gas constant if
kilopascals.
liters;
mole,
K;
Example: The volume of a geometric figure varies jointly with the square of the radius of the base and
the height.
Write an equation for the volume.
Estimate the constant of variation if
Chapter 9.2: Graphing Simple Rational Functions
Graphing a Simple Rational Function:
;
in;
in.
Rational Functions in the form
Called a
Horizontal asymptote?
Vertical asymptote?
Domain?
Range?
Branches:
Rational Functions in the form
Asymptotes:
To graph
Example: Graph
. State the domain and range.
Example: Graph
. State the domain and range.
Rational Functions in the form
Asymptotes:
Example: Graph
. State the domain and range.
Example: The senior class is sponsoring a dinner. The cost of catering the dinner is $9.95 per person
plus an $18 delivery charge.
Write a model that gives the average cost per person.
Graph the model and use it to estimate the number of people needed to lower the cost to $11 per
person.
Describe what happens to the average cost per person as the number increases.
Example: The speed of sound can be modeled by
ft/sec, where F is the temperature in
degrees Fahrenheit.
Write a model that gives the time it takes you to hear thunder a mile away.
Graph the model and use it to estimate how long it takes you to
hear the thunder a mile away if it is 75 degrees.
Describe what happens to the length of time it takes for the
thunder to reach your ears as the temperature decreases.
Chapter 9.3: Graphing General Rational Functions
Graphs of Rational Functions:
( )
( )
( )
Example: Graph
. State the domain and range.
Example: Graph
. State the domain and range.
Example: Graph
Example: Graph
.
.
Example: Graph
.
Example: Graph
.
Example: A frozen yogurt cone has a volume of 10 cubic inches.
The surface area S of a cone excluding the base is
, where r is the radius of the base and
√
h is the height.
Find the dimensions of the cone that has this volume and the smallest surface area possible.
Compare your results with the dimensions of an actual cone, which has a radius of 1.26 in. and a
height of 6 in.
Example: A silo is to be built in the shape of a cylinder with a volume of 100,000 cubic feet. Find the
dimensions of the silo that use the least amount of material. Include the top and bottom surfaces of the
silo.
Chapter 9.4: Multiplying and Dividing Rational Expressions
Simplifying Rational Expressions:
Example:
Simplify:
Example: Multiply
Simplify:
(
)
(
)
Example: Divide
(
)
Example: Simplify
(
)
(
)
Example: The diagram shows a simplified version of a robotic dog.
Use the diagram to write a model for the ratio of the volume to
the cross-sectional area of the dog’s feet.
Use the ratio found, find the value of the ratio for two robots, one with a tail 10 centimeters long
and a second with a tail 15 centimeters long.
Chapter 9.5: Addition, Subtraction, and Complex Fractions
Example: Perform the indicated operation.
Example: Josh drive 42 miles and then took the train. The entire trip was 172 miles. The average speed
of the train was 35 mi/h faster than the average speed of the car. Let x equal the average speed of the car
and y equal the total time traveled. Then
is the time the car traveled and
is the time
the train traveled.
Graph each model. What is the time Josh traveled in the car if his rate is 35 mi/h? What is his
time on the train if his car rate is 35 mi/h?
Write a model that shows the total time it takes for the trip. Graph the model. If the car’s
average speed is 50 mi/h, how long does the trip take?
Example: Given the following rational functions.
and
(
)
(
)
Graph each function. For what values of x is
a maximum?
?
Graph a model that shows the sum of the functions. For what value of x is this function a
maximum?
Complex Fraction
Example: Simplify:
Example: The focal length f (in centimeters) of a curved mirror is
distance from the mirror and
is the object’s
is the image’s distance from the mirror. Simplify the complex fraction.
Chapter 9.6: Solving Rational Equations
Solving a Rational Equation:
Example: Solve:
, where
Example: You have 1.4 liters of an acid solution whose acid concentration is 2.1 moles per liter. You
want to dilute the solution with water so that its acid concentration is 1.5 moles per liter. How much
water should you add to the solution?
Example: In economics, an increasing supply curve means that as prices increase, sellers usually
increase production. A decreasing demand curve means that as price increase, consumers buy less.
Suppose that a market situation is modeled by the following equations:
Supply:
Demand:
Use the Intersection feature of a graphing calculator to determine the equilibrium price – the price at
which the supply equals the demand.