This document discusses rational functions and provides examples of representing rational functions through tables of values, graphs, and equations. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not the zero function. Examples are given of using rational functions to model speed as a function of time for running a 100-meter dash and calculating winning percentages in a basketball league.

1. RATIONAL FUNCTIONS
Representations of Rational Functions
www.slideshare.net/reycastro1
General Mathematics

2. RATIONAL FUNCTIONS
› LEARNING OUTCOMES:
› Able to represent a rational function
through its table of values, graphs and
equation, and solve problems involving
rational functions.

3. RATIONAL FUNCTION
A function of the form of 𝒇 𝒙 =
𝒑(𝒙)
𝒒(𝒙)
where
𝒑(𝒙) and 𝒒(𝒙) are polynomials, and 𝒒(𝒙) is
not the zero functions. The domain of f(x) is all
values of x where q(x) ≠ 0.
Example: 𝒇(𝒙) =
𝒙 𝟐+𝟐𝒙+𝟑
𝒙+𝟏
or 𝒚 =
𝒙 𝟐+𝟐𝒙+𝟑
𝒙+𝟏

4. Rational Function Model
Average speed (or velocity) can be computed
by the formula 𝒔 =
𝒅
𝒕
.
Example: Consider a 100-meter track used
for foot races. The speed of a runner can be
computed by taking the time it will take him
to run the track and applying it to the
formula 𝒔 =
𝟏𝟎𝟎
𝒕
. Since the distance is fixed at
100 meters.

5. Rational Function Model
Example 1: Represent the speed of a
runner as a function of the time it takes to
run 100 meters.

6. Rational Function Model
Example 1: Represent the speed of a
runner as a function of the time it takes to
run 100 meters.
Let x represent the time, then the speed
𝒔 =
𝟏𝟎𝟎
𝒕
is 𝒔 𝒙 =
𝟏𝟎𝟎
𝒙

7. Rational Function Model
Example 2: Construct a table of values for
the speed of a runner against different run

8. Rational Function Model
Example 2: Construct a table of values for
the speed of a runner against different run
A table of values can help as determine the
behavior of a function as the variable x
changes.

9. Rational Function Model
Let x be the run time and s(x) be the speed
of the runner in m/s.
x 10 12 14 16 18 20
s(x)
The current world record (as of October
2015) for the 100-meter dash is 9.58
seconds set by the Jamaican Usain Bolt in
2009.

10. Rational Function Model
Example 3: Plot the points in the table of
values on the Cartesian plane. Determine if
the points of 𝒔 𝒙 =
𝟏𝟎𝟎
𝒙
follow a smooth
curve or straight line.
Assign point on the Cartesian plane;
S = {(10,10),(12,8.33),(14,7.14),(16,6.25),(18,5.56),(20,5)}

11. Rational Function Model
S = {(10,10),(12,8.33),(14,7.14),(16,6.25),(18,5.56),(20,5)}

12. Rational Function Model
S = {(10,10),(12,8.33),(14,7.14),(16,6.25),(18,5.56),(20,5)}
For the 100-meter dash scenario, we have
constructed a function of speed against time,
and represented our function with a table of
values and a graph.

13. Rational Function Model
Example 4: In an inter-barangay basketball league.
The team from Barangay Sto. NiÑo has won 12 out of
25 games, a winning percentage of 48%. We have
seen that they need to win 8 games consecutively to
raise their percentage to at least 60%. What will be
their winning percentage if they win
(a) 10 games in a row?
(b) 15? 20? 30? 50? 100 games? Can they reach a
100% winning percentage?

14. Rational Function Model
Solution: Let x be the number of wins and
percentage p is a function of the number of wins.
𝒑 𝒙 =
𝟏𝟐 + 𝒙
𝟐𝟓 + 𝒙
Construct a table of values for p(x)
x 8 10 15 20 30 50 100
p(x) 60%

15. Rational Function Model
Example 5: Ten goats were set loose in an island and
their population growth can be approximated by the
function
𝑃 𝑡 =
60(𝑡 + 1)
𝑡 + 6
where P represents the goat population in year t since
they were set loose. Recall that the symbol . denotes
the greatest integer function.
(a) How many goats will there be after 5 years?
(b) What is the maximum goat population that the island
can support?

16. Try this!
1. Construct a table of values for the function
𝑓 𝑥 =
𝑥−3
𝑥+4
for −6 ≤ 𝑥 ≤ 2 , x taking on integer
values. Identify values of x where the function will
be undefined. Plot the points corresponding to
values in the table. Connect these points with a
smooth curve.

17. THANK YOU
www.slideshare.net/reycastro1
@reylkastro2
reylkastro