Chapter 2: Rational Function

Objectives:
• Review polynomial function;
• Present real life examples of
rational function; and
• Define rational function.

What is a Polynomial
Function?

All Rights Reserved
What’s on your mind?
Polynomial
Not Polynomial
Polynomial
Not Polynomial
Not Polynomial
Not Polynomial
Not Polynomial
Not Polynomial
Not Polynomial
Polynomial
With x on denominator
Imaginary coefficient
Negative exponent
With x inside the radical
Fractional exponent

1. …there is no imaginary coefficient on any
term.
2. …there is no negative exponent on terms with
x on the numerator.
3. …no terms with x on the denominator.
4. …no fractional exponent or no x variable
inside the radical sign.

1. The local barangay received a budget of PhP 100,000
to provide medical check ups for the children in the
barangay.
a. If the amount is to be allotted equally among all the
children in the barangay. Write an equation
representing the relationship of the allotted amount
per child (y-variable) versus the total number of
children (x-variable)
b. A philanthropist wants to supplement the budget
allotted for each child by providing an additional PhP
750 for each child in the barangay. If g(x) represents
this new amount allotted for each child, construct a
function representing this relationship.

1. An object is to travel a distance of 10 meters.
Express velocity v as a function v(t) of travel
time t, in seconds.
2. 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 off answers to
three decimal places. Use the table to sketch
a graph and interpret the result.

Situation 1: The budget of a university
organization is split evenly among its
various committees. If they have a budget
of P60,000:
a. Construct a function M(n) which would give
the amount of money each of the n number
of committees would receive.
b. If the organization has eight committees,
how much would each committee have?

Situation 2: Let C(t) = be the function that
describes the concentration of a certain
medication in the bloodstream over time.
a. What is C(0)? Why is that so?
b. Construct a table of values for when t is
equal to 0,1,2,3,4, and 5.
c. Interpret your answers in relation to drug
concentration.

1. A fence is to enclose a rectangular vegetable
farm with an area of 400 sq. meters. If x is the
length of one side of this fence, find a function
P(x) representing the perimeter of the fencing
material required. Find the perimeter of the fence
if one side has length 10 meters, 50 meters.
2. Suppose the amount of bacteria growing in a
Petri dish is represented by the function
where t is in hours and b(t) is in millions. Evaluate
the function at t = 1, 2, , 5 and 10. Interpret the
obtained values.

Objectives:
• Distinguish rational functions,
rational equations and rational
inequalities.

A rational expression is an expression
that can be written as a ratio of two
polynomials.

Determine whether the given is a rational
equation, rational function, rational inequality or None
of these.

Objectives:
• Solve rational equations; and
• Solve problems involving
rational equations.

Warm Up!
5(x + 1/5)
12(m + 1/12)
3x(4/x + 4)
10a(1/2a + 3/5a)
12m(2/3m - 5/2m)
(x + 2)(3 - 4/(x+2))
5x + 1
12m + 1
12 + 12x
11
- 22
3x + 2

To solve a rational equation:
(a) Eliminate denominators by multiplying
each term of the equation by the least
common denominator.
(b) Note that eliminating denominators
may introduce extraneous solutions.
Check the solutions of the transformed
equations with the original equation.

Answer:
• The LCD of all the denominators is 10x. Multiply
both sides of the equation by 10x and solve the
resolving equation.

Answer:
• The LCD of all the denominators is (x + 2)(x – 2).
Multiply both sides of the equation by (x + 2)(x – 2)
and solve the resolving equation.

Bring It On!
Solve for x.
1.
2.

Let’s Try This!
1. In an inter-barangay basketball league, the team
from Barangay Culiat has won 12 out of 25 games,
a winning percentage of 48%. How many games
should they win in a row to improve their win
percentage to 60%
Solution. Let x represent the number of games that they need
to win to raise their percentage to 60%. The team has already
won 12 out of their 25 games. If they win x games in a row to
increase their percentage to 60%, then they would have played
12+x games out of their 25+x games. The equation is
Multiply 25+x to both sides of the equation and solve the
resulting equation.

Let’s Try This!
• Jens walks 5 kilometers from his house to Quiapo to
buy a new bike which he uses to return home. He
averaged 10 kilometers faster on his bike than on
foot. If his total trip took 1 hour and 20 minutes, what
is his walking speed in kph? Use the formula v = d/t.

Objectives:
• Solve rational inequalities.

Warm Up:
x = -1 x = 0
x = 0 x = 5
x = -1, 1 x = -2/3
x = 0 x = - 2
x = -1, -2 x = -3, 3

Interval and Set Notation

To Solve Rational Inequalities
1. Solve for x – intercept:
Let the numerator equal to zero
and solve for x.
(use solid dot for x intercept
because it is included on the
graph)
2. Solve for Vertical Asymptote:
Let the denominator equal to
zero and solve for x.
(use hollow dot for vertical
asymptote because it is not
included on the graph)

To Solve Rational Inequalities

Solve the inequality
1.
2.
2x
x + 1
 1
3
x – 2
< 1
x

Objectives:
• Graph rational functions.

Mechanics of the game:
1. The game should be played by three groups. Each group
should have a team leader to lead the group.
2. The game starts by presenting the first question in random.
The first team to answer the question correctly will be given a
chance to choose the category of the next question and will be
given a time to answer depending on the value of the
question. Once the time is over, the other teams will be given
a chance to answer the said question.
3. Two consecutive chances will be given to every team who can
answer the questions simultaneously and on the third time,
the team will be given a chance to choose “play” or “pass”. If
they choose play, the team can go on in answering the next
questions but once they choose pass, the question will be
given to the other teams.

4. If the team answered the three questions simultaneously, a
breaker question will be given by the game master to give
chance to other teams to answer the question.
5. If none of the team answered the question correctly, a tie-
breaker question will be given by the game master.
6. Once all the questions are revealed, a final jeopardy will be
given by the game master.
7. A team with a highest point will be declared a winner.
8. The accumulated points earned by each team corresponds
to the points to be given to each member of the team with 6
points higher for the representatives.
9. Time allocation for every question:
P100 to P300 ------------------------ 20 seconds
P400 to P500 ------------------------ 40 seconds
Final Jeopardy (P1000) ----------- Open time

Graphing Rational Functions:
1.Rewrite the rational function as a
factored form.
2.Find the Intercepts:
a. x-intercept.
Let y equal to zero and solve for x.
b. y-intercept.
Let x equal to zero and solve for y.

3. Find the Asymptotes:
a. Vertical Asymptote.
Let the denominator equal to zero and solve for x.
b. Horizontal Asymptote.
For rational function f(x) = axn + bxn-1+…,
bxm + cxm-1+…
 If n < m, the horizontal asymptote is y = 0.
 If n = m, the horizontal asymptote is a/b.
 If n > m, there is no horizontal asymptote
c. Oblique Asymptote.
Divide the numerator by the denominator.
The quotient is the oblique asymptote of the
rational function.

4.Plot the points on the coordinate plane.
5.Using the sign analysis test/table of signs,
determine where the graph is passing for
every region of the coordinate plane. If the
outcome of the sign on a particular region is
positive, shade the region below the x-axis.
If the outcome is negative, shade the region
above x-axis.
6.Determine the path of the curve on the
coordinate plane and analyze the end
behaviour of the graph with respect to the
asymptotes.
7.Sketch the graph using a smooth curve.

Example:
1.Sketch the graph of y = x2 – 9.
x2 – 4x + 3
1st Step: Write in factored form:
y = (x – 3)(x + 3)
(x – 3)(x - 1)
y = x + 3
x - 1

2nd Step: Find the intercepts:
x-intercept: y = (x + 3)
(x - 1)
y-intercept: y = (x + 3)
(x - 1)
0 = x + 3
x - 1
0 = x + 3
x = - 3
y = 0 + 3
0 - 1
y = - 3
“Let the y or the numerator
equal to zero and solve for x.”
“Let x equal to zero
and solve for y.”

3rd Step: Find the asymptote:
Vertical Asymptote:
y = (x + 3)
(x - 1)
Horizontal Asymptote:
Oblique Asymptote:
0 = x - 1
x = 1
y = 1
“Let the denominator equal
to zero and solve for x.”
“Since the degree of the numerator is
equal to the degree of the denominator,
therefore, The horizontal asymptote is a/b
or y = 1.
“None”

4th Step: Plot the points on a number line:
Region 1
x ≤ -3
Region 2
-3 ≤ x < 1
Region 3
x > 1

5th Step: Analyze the path of the graph:
Intervals x ≤ -3 -3 ≤ x < 1 x > 1
Test Points x = -4 x = 0 x = 2
(x + 3) - + +
(x – 1) - - +
(x + 3)
(x – 1) (+) (-) (+)
The graph is passing
above
the x-axis
below
the x-axis
above
x-axis
Shade the region
below the x-axis
Shade the region
above the x-axis
Shade the region
below the x-axis

6th Step: Sketch the graph:

Sketch the graph of the following:
1. y = 2x – 4 2. y = x2 + 3x - 10
2x2 – 8 x2 – 25
3. y = 6x + 12 4. y = x2 – 4x + 4
x2 + 4x + 4 x2 - 9

Chapter 2: Rational Function

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Chapter 2: Rational Function