Graphs of Rational Function

1. GRAPHING RATIONAL
FUNCTIONS
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General Mathematics

2. GRAPHING RATIONAL FUNCTIONS
› LEARNING OUTCOMES:
› Able to find the domain and range,
intercepts, zeros, asymptotes of
rational functions graph rational
functions and solve problems involving
rational functions.

3. (a) The domain of a function is the set of all values
that the variable x can take.
(b) The range of a function is the set of all values
that f(x) can take.
(c) The zeros of a function are the values of x which
make the function zero. The real numbered zeros
are also x-intercepts of the graph of the function.
(d) The y-intercept is the function value when x=0.
Properties of Functions

4. Example 1: Consider the function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
(a) Find its domain, (b) intercepts, (c) sketch its
graph and (d) determine its range.
GRAPHING RATIONAL FUNCTIONS

5. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
GRAPHING RATIONAL FUNCTIONS

6. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
Solution. (a) The domain of is 𝒙 ∈ ℝ|𝒙 ≠ −𝟐 ,
(b) The x-intercept of f(x) is 2 and its y-intercept
is -1.
GRAPHING RATIONAL FUNCTIONS

7. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
(c) To sketch the graph of f(x), let us look at what
happens to the graph near the values of x which
make the denominator undefined.
 When x=-2, f(x) is undefined.
 Values of x close to -2; x < -2, denoted -2-.
GRAPHING RATIONAL FUNCTIONS

8. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
 Values of x close to -2; x < -2, denoted −𝟐−.
 Notation: f(x) → +∞ as x → −𝟐−.
GRAPHING RATIONAL FUNCTIONS

9. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
 Values of x approaching, −𝟐+.
 Notation: f(x) → −∞ as x → −𝟐+.
GRAPHING RATIONAL FUNCTIONS

10. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
 As f(x) → +∞ then x → −𝟐−.
 And f(x) → −∞ then x → −𝟐+.
GRAPHING RATIONAL FUNCTIONS
Definition:
The vertical line x=a is a vertical
asymptote of a function f if the graph of
f either increases or decreases without
bound as the x-values approach a from
the right to left.

11. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
 Table of values for f(x) as x→ +∞.
 Table of values for f(x) as x→ −∞.
GRAPHING RATIONAL FUNCTIONS

12. Example 1: function 𝒇 𝒙 =
𝒙−𝟐
𝒙+𝟐
.
 As x→ +∞ then f(x) → 𝟏−.
 And x→ −∞ then x → 𝟏+.
GRAPHING RATIONAL FUNCTIONS
Definition:
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 𝒙 → +∞ 𝒐𝒓 𝒙 → −∞ .

13. Construct a table of signs to determine the sign
of the function on the intervals determined by
the zeros and the vertical asymptotes.
GRAPHING RATIONAL FUNCTIONS

14. Plot the zeros, the y-intercept, and the
asymptotes.
GRAPHING RATIONAL FUNCTIONS
The actual sketch of the graph of 𝑦 =
𝑥−2
𝑥+2
(d) The range of f(x) is −∞, 𝟏 ∪ 𝟏, +∞ .

15. Horizontal asymptote of a rational function.
Let n be the degree of the numerator and m
be the degree of the denominator.
• If n<m, the horizontal asymptote is y=0.
• If n=m, the horizontal asymptote is 𝐲 =
𝒂
𝒃
.
• If n>m, there is no horizontal asymptote.
HORIZONTAL ASYMPTOTE

16. Horizontal asymptote of a rational function.
Example 2: Find the horizontal asymptote of
𝒇 𝒙 =
𝟒𝒙 𝟐+𝟒𝒙+𝟏
𝒙 𝟐+𝟑𝒙+𝟐
Thus, the horizontal asymptote is y=4.

17. Horizontal asymptote of a rational function.
Example 3: Find the horizontal asymptote of
𝒇 𝒙 =
𝟐𝒙 𝟐−𝟓
𝟑𝒙 𝟐+𝒙−𝟕
Thus, the horizontal asymptote is 𝒚 =
𝟐
𝟑
.

18. Horizontal asymptote of a rational function.
Example 4: Find the horizontal asymptote of
𝒇 𝒙 =
𝟑𝒙 + 𝟒
𝟐𝒙 𝟐 + 𝟑𝒙 + 𝟏
Thus, the horizontal asymptote is ______.

19. Horizontal asymptote of a rational function.
Example 5: Find the horizontal asymptote of
𝒇 𝒙 =
𝟒𝒙 𝟑
− 𝟏
𝟑𝒙 𝟐 + 𝟐𝒙 − 𝟓
Thus, the horizontal asymptote is ______.

20. Horizontal asymptote of a rational function.
Example 6: Sketch the graph of
𝒇 𝒙 =
𝟑𝒙 𝟐−𝟖𝒙−𝟑
𝟐𝒙 𝟐+𝟕𝒙−𝟒
. Find its domain
and range.

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