This document discusses polynomial and rational functions. A polynomial function of degree n can be written as p(x) = anxn + an-1xn-1 + ... + a1x + a0, where the coefficients a0, a1, ..., an are real numbers and an ≠ 0. A rational function is defined as f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions and q(x) ≠ 0. Several examples are provided of using rational functions to model real-world situations involving distance-time, concentration of a drug over time, and allocating a budget amount per child based on the total number of children.

2. Polynomial Function of P of Degree
in is function that can be written in
form 𝑝 𝑥 = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+
𝑎𝑛−2𝑥𝑛−2
+ ⋯ + 𝑎1𝑥 + 𝑎0 , where
𝑎0, 𝑎1, … , 𝑎𝑛 ∈ ℝ, 𝑎𝑛 ≠ 0, and n is a
positive integer.

3. 𝑝 𝑥 = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+
𝑎𝑛−1𝑥𝑛−2
+ ⋯ + 𝑎1𝑥 + 𝑎0

4. 𝑝 𝑥 = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+
𝑎𝑛−2𝑥𝑛−2
+ ⋯ + 𝑎1𝑥 + 𝑎0

5. Rational function of the form 𝑓 𝑥 =
𝑝 𝑥
𝑞(𝑥)
where 𝑝 𝑥 and 𝑞 𝑥 are
polynomial functions and 𝑞 𝑥 is not
the zero function ( 𝑞(𝑥) ≠ 0 ). The
domain of 𝑓 𝑥 is the set of all values
of x where 𝑞(𝑥) ≠ 0.

6. Representing Real-Life
Situations Using
Rational Functions
General Mathematics

7. At the end of the lesson, the
students are expected to:
a. demonstrates understanding
of key concepts of rational
functions; and
b. accurately solve real-life
problems involving rational functions

8. The local barangay received a budget
of P100 000.00 to provide medical
check-ups for the children in the
barangay. The amount is to be
allotted equally among all the
childreninthebarangay.

9. STEP 1:
Write an equation representing of the
allotted amount per child (y-variable)
versus the total number of children (x-
variable)
𝒚 =
𝟏𝟎𝟎𝟎𝟎𝟎
𝒙

10. Fill up the table below with different allotment
for different values for the number of children.
𝒚 =
𝟏𝟎𝟎𝟎𝟎𝟎
𝒙
10 20 50 100 200
Allocated
amount , y
10 000 5 000 2 000 1 000 500

11. An object is to travel a
distance of 10 meters. Express
velocity v(t) of travel time t, in
seconds.

12. Following the table of values below, show v for
various values of t.
1 2 3 4 5
v (meters
per second)
10 5 2.5 2 1

13. The function of 𝐯 𝐭 =
𝟏𝟎𝟎𝟎𝟎𝟎
𝒙
can represent v as a function
of t.

15. The local barangay received a budget of P100
000.00 to provide medical check-ups for the
children in the barangay. The amount is to be
allotted equally among all the children in the
barangay. A philanthropist wants to
supplement the budget allotted for each child
by providing an additional P750 each child in
thebarangay.

16. In an organ pipe, the frequency f of
vibration of air is inversely proportional to
the length L of the pipe. Suppose that the
frequency of vibration in a 10-foot-pipe is
54 vibrations per second. Express f as a
functionofL.

17. The distance from Manila to Baguio is around 250
kilometers.
a. How long will take you to get to Baguio if your
average speed is 25 kilometers per hour? 40
kilometers perhour?50kilometers perhour?
b. Construct a function (s), where s the speed of
travel that describes the time it takes to drive from
ManilatoBaguio

18. Suppose that 𝒄 𝒕 =
𝟓𝒕
𝒕𝟐+𝟏
(in mg/mL) represents the
concentration of a drug in a patient’s bloodstream t
hours after the drug was administered. Construct a
table of values for c(t) for t=1,2,5,10. Round your
answers to three decimal places. Use the table to
sketchagraphandinterprettheresults.