Download INTRO TO RATIONAL FUNCTIONS Light It Up is the newest

INTRO TO RATIONAL FUNCTIONS
Light It Up is the newest attraction at the Springfield Fair. In this game, a laser pointer slides along a string at a height of
1.5 m from the ground. A 10-cm platform is positioned 25 cm from a wall, and a mirror is placed on top of the platform,
parallel with the ground. The object is to slide the laser pointer so that when the beam is reflected off the mirror, it will
hit a target that is 1 m off the ground.A player is given just one attempt to hit the target. If she does, she then tries to hit
a target that is 3 m off the ground. And if she hits that one, too, she tries for a target thatis 10 m off the ground. If she
hits all three, she wins the Grand Prize: a DVD player and a DVD copy of Itchy and Scratchy’s Greatest Hits.Bart and Lisa
have played the game several times, yet they haven’t been able to hit the target. Lisa is sure that there must be a
mathematical equation that would allow them to determine the correct position for the laser pointer. There issuch an
equation, it is a special kind known as a rational function Before you can help Lisa and Bart win this game, you must
first understand the behavior of these functions
CLIMIBING THE WALL
1. TheClimbing Wall is a popular event at the Springfield Fair. The wall is 100 meters high. Bart can climb at a rate
of 8 meters per minute, but other participants climb at different speeds.
a.Complete the table to show that the time it takes to reach the top of wall depends on the climber’s speed .
b.
Determine an equation that describes the table of data
c.
Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in
negative values and solve for cost)
d. Describe the domain and range. Are there any limiting values of the function?
Activity taken from
© 2008National Council of Teachers of Mathematics http://illuminations.nctm.org
TRIP TO THE FAIR
2. Ms. Crabapple, a teacher at Springfield Elementary, is sponsoring a school trip to the fair. The total cost for
transportation, parking and entrance fees is $1200. The total cost will be divided among all the students who go
on the trip.
a. Complete the table to show how much each student must pay to cover expenses.
b.
Determine an equation that describes the table of data
c.
Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in
negative values and solve for cost)
d.
Describe the domain and range. Are there any limiting values of the function?
e.
How are the equations of questions 1 and 2 similar? Different?
f.
How are the graphs of questions 1 and 2 similar? Different?
Activity taken from
© 2008National Council of Teachers of Mathematics http://illuminations.nctm.org
The functions in Questions 1 and 2 are known as rational functions. Rational functions are quotients of two
polynomials functions. In other words f(x)=
p( x)
, where p(x) and q(z) are polynomials and q≠0.
q( x)
Why can q(x) ≠0?
TRIP TO THE FAIR PART TWO
Ms. Crabapple was just informed that the new attraction, Light It Up will cost an additional $5 per student.
a. Complete the table to reflect the change to the amount each student must pay to cover expenses .
b. Write an equation that describes the table.
c.
Some rational functions can be written in the form of y 
a
 k , where a,h, and k are constants. If needed,
xh
rewrite your equation from part b in this form.
d. Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in
negative values and solve for cost)
e. What is your domain and range and how do they relate to the problem?
Activity taken from
© 2008National Council of Teachers of Mathematics http://illuminations.nctm.org
VARIATIONS
Rational functions are a type of inverse variation.
There are two main types of variations; direct and inverse
DIRECT
A direct variation is a relationship in which as one variable increases , the other variable increases proportionally or as
one variable decreases the other variable decreases proportionally.Basically the two variables ‘react’ in the same manner.
A direct variation equation is linear and can be written in the form of y=mx+b where b=0 and the constant of variation
is the slope. The graph of a direct variation always passes through the origin(this is what MAKES it proportional).
The general form of a direct variation is y=kx
k represents the constant of variation
INVERSE
An inverse variation describes a situation in which one quantity increases as the other decreases proportionally (like the
amount of time you are on facebook vs your algebra II grade) . The two variables ‘react’ oppositely.
The general form of a direct variation y = k/x where k≠0 k represents the constant of variation
At times you may see the term joint variation. This represents a direct relationship among 3 variables where the
equation would be y=kxz where k is the constant of cariations
COMBINED
A combined variation is a relationship that contains both direct and inverse variation. Quantities that vary directly appear
in the numerator and quantities that vary inversely are in the denominator
EXAMPLES
a. Given y varies directly as x, and y = 14 when x = 3.5. Write
and graph the direct variation function.
y=kx
14=k(3.5)
k=14/3.5
k=4
y=4x
b.Given y varies inversely as x, and y=2 when x=5. Write and
graph the direct variation function.
y=k/x
2=k/5
k=6
y=6/x
Identify the type of variation and then write the equation.
COMPLETE THE GOOGLE SURVEY! LINK IN HAIKU!
Activity taken from
© 2008National Council of Teachers of Mathematics http://illuminations.nctm.org