Malapitan_Q1WEEK4

Quarter 1 Week 4
Determining intercepts, zeroes, asymptotes of rational functions
Mary Ashley S. Malapitan

11- Artemis
Learning Competency: Determine the intercepts, zeroes, and asymptotes
of rational functions
INTERCEPTS AND ZEROES
The y- intercept of the graph of a

rational function (fx) if it exists, occurs
at f(0), provided that f(x) is defined
at x = 0. To find y-intercept simply
evaluate the function at x = 0.
The x- intercept of the graph of a
rational function f(x), if it exists,
occurs at the zeros of the numerator
that are not zeros of the denominators.
To find x – intercept equate the function
to 0.
The zeroes of a function are the
values of x which make the function zero.
The numbered zeroes are also x-
intercepts of the graph of the function.
Directions: Identify the x and y-intercepts o

2
2x
1. f ( x )= 2
3 x +1
2 x−4
2. f ( x )=
x−6
2 x−3
3. f ( x )= 2
3 x +1
ASYMPTOTES
An asymptote is an imaginary line to which a graph gets closer and closer as
the x or y increases or decreases its value without limit.
VERTICAL ASYMPTOTE. The vertical line 𝑥 = 𝑎 is a vertical asymptote of a
approach 𝒂 from the right or left. To determine the vertical asymptote of a

function f if the graph increases or decreases without bound as the x values
rational function, first reduce the given function to simplest form then find
the zeroes of the denominator that are not zeros of the numerator.
HORIZONTAL ASYMPTOTE. The horizontal line y=b is a horizontal
asymptote of the function f if f(x) gets closer to b as x increases or decreases
without bound. We can determine horizontal asymptote arithmetically by
comparing the degree of the leading coefficient of the numerator and
denominator of the function. To determine the horizontal asymptote of
a rational function, compare the degree of the numerator n and the
degree of the denominator d.
• If n < d, the horizontal asymptote is y= 0
the numerator a, to the leading coefficient of the denominator b. That is 𝑦 =

• If n = d, the horizontal asymptote y is the ratio of the leading coefficient of
a
b
• If n > d, there is no horizontal asymptote.
Directions: Give what is asked.

2 x−4
4. The horizontal asymptote of f ( x )=
x−6
x+1
5. The vertical asymptote of f ( x )= 2
x −1
3 x+ 8
6. The vertical and horizontal asymptote of f ( x )= 2
x +1
Provide the intercepts and asymptotes of the following:

2
x + x−6
7. f ( x )= 2
−4 x −16 x−12
3 2
x −x −6 x
8. f ( x )= 2
−3 x −3 x+ 18
1
9. f ( x )= 2
3 x + 3 x +18
3 x +6
10. f ( x )=
x−1
ANSWER KEY
1. x-intercept:
2
2x
f ( x )= 2
3 x +1
0=2 x ; x=0
2
y-intercept:
f ( 0 )=2 ¿ ¿
2. x-intercept:
2 x−4
f ( x )=
x−6
0=2 x −4 ; 2 x=4
y-intercept:
2 ( 0 )−4 −4 2
f (0)= ; f ( 0 )= ; f (0)=
0−6 −6 3
3. x-intercept:
2 x−3
f ( x )= 2
3 x +1
3
0=2 x −3; 3=2 x ; x=
2
y-intercept:
2 ( 0 )−3 −3
f (0)= ; f ( 0 )= ; f ( 0 )=−3
3¿ ¿ 1
2 x−4
4. The horizontal asymptote of f ( x )=
x−6
a 2
n=d ; therefore, y= = ; y=2
b 1
x+1
5. The vertical asymptote of f ( x )= 2
x −1
2 2
x −1=0; x =1
x=± 1, however, x= -1 is a zero of the numerator.
x=1
3 x +8
6. The vertical and horizontal asymptote of f ( x )= 3
x +27
3 3
x +27=0 ; x =−27
x=√−27
3
x=−3
n<d ; therefore, y=0
7.
Vertical A. : x
−1
Horz. A.: y
4
X-intercept: 
1
y-intercept:
2
8.
Vertical A. : x, x
Horz. A.: None
X-intercepts: , , 
y-intercept: 0
9.
Vertical A. : x, x
Horz. A. : y
X-intercepts: None
y-intercept: None
10.
Vertical A. : x1
Horz. A. : y3
X-intercepts: -2
y-intercept: -6
References:
https://math.libretexts.org/Courses/Monroe_Community_College/
MTH_165_College_Algebra_MTH_175_Precalculus/
03%3A_Polynomial_and_Rational_Functions/3.9%3A_Rational_Functions/3.9e
%3A_Exercises_-_Rational_Functions
https://www.scribd.com/document/480611662/GenMath11-Q1-Mod9-
intercepts-zeroes-and-asymptotes-of-functions-08082020-pdf
https://www.chino.k12.ca.us/site/handlers/filedownload.ashx?
moduleinstanceid=5435&dataid=2417&FileName=UNIT
%204%20WORKSHEET%2012%20RATIONAL%20ASYMPTOTES.pdf

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Quarter 1 Week 4

Determining intercepts, zeroes, asymptotes of rational functions

Mary Ashley S. Malapitan

INTERCEPTS AND ZEROES

The y- intercept of the graph of a

Directions: Identify the x and y-intercepts o

approach 𝒂 from the right or left. To determine the vertical asymptote of a

the numerator a, to the leading coefficient of the denominator b. That is 𝑦 =

Directions: Give what is asked.

Provide the intercepts and asymptotes of the following:

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