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Rational Function
function 𝑓(𝑥) is the quotient of two
polynomial functions 𝑎(𝑥) and 𝑏(𝑥), where 𝑏 is
nonzero.
 A rational
𝑎 𝑥
𝑓 𝑥 =
𝑏 𝑥
 The domain of a rational function is all real
numbers excluding those values for which
𝑏 𝑥 = 0, or the zeroes of 𝑏 𝑥 .
Rational Functions
1
𝑥

𝑓 𝑥 =

This is not piecewise.

The lines representing those
values are called
asymptotes.
Graph
𝑓
𝑥 =
1
𝑥−2
Asymptotes
line 𝑥 = 𝑐 is a vertical asymptote of 𝑓
if 𝑓 𝑥 → ±∞ as 𝑥 → 𝑐.
 The
Asymptotes
line 𝑦 = 𝑏 is a horizontal asymptote of
𝑓 if 𝑓 𝑥 → 𝐛 as 𝑥 → ±∞.
 The
Find the Domain and the Asymptotes
𝑓
𝑥 =
𝑥+4
𝑥−3
Check with a graph
Find the Domain and the Asymptotes
𝑔
𝑥 =
8𝑥 2 +5
4𝑥 2 +1
Check with a Graph
Find the Domain and the Asymptotes
𝑚
ℎ
𝑥 =
𝑥 =
15𝑥+3
𝑥+5
𝑥 2 −𝑥−6
𝑥+4
Vertical Asymptotes
 Vertical
asymptotes can only exist where the
domain of a function is discontinuous.
 But
just because a function is discontinuous
a particular value of 𝑥 doesn’t mean it will
form an asymptote, so check.
Holes
 Given
the polynomial function:
𝑥2 − 4
𝑓 𝑥 = 2
𝑥 − 3𝑥 − 10
What is the domain of this?
Holes
𝑥2 − 4
𝑓 𝑥 = 2
𝑥 − 3𝑥 − 10
Horizontal Asymptotes
Let 𝑓 be a rational function defined as:
𝑎 𝑥
𝑓 𝑥 =
, 𝑏(𝑥) ≠ 0
𝑏 𝑥
𝑎(𝑥) and 𝑏 𝑥 are polynomials with no common factors. Let 𝑎(𝑥)
have degree 𝑛 and 𝑏(𝑥) have degree 𝑚
 The graph may have 1 or 0 horizontal asymptotes using these
guidelines:
 If 𝑛 < 𝑚, the horizontal asymptote is 𝑦 = 0.


𝑎
If 𝑛 = 𝑚, the horizontal asymptote is 𝑦 = 𝑏 𝑛
𝑚

If 𝑛 > 𝑚, there is no horizontal asymptote.
𝑥-Intercepts and 𝑦-Intercepts
𝒙-intercepts of 𝑓(𝑥) occur at the zeroes
of 𝑎 𝑥 .
 All
 Why?
 The
𝑦-intercepts occur at 𝑓(0).
Find the Domain, Asymptotes, and 𝑥and 𝑦-intercepts
ℎ
𝑛
𝑥 =
2
𝑥 2 +2𝑥−3
𝑥 =
𝑥 2 −4
5𝑥 2 −5
Do Now
 Find
all asymptotes, holes, and intercepts
for:
𝑥 2 + 5𝑥 − 50
𝑓 𝑥 = 2
𝑥 + 𝑥 − 30
Do Now
 Write
a procedure for determining the slant
asymptotes of rational equations.
 Use
that procedure to find all asymptotes
for:
3𝑥 3 − 7𝑥 2 − 22𝑥 + 8
𝑓 𝑥 =
𝑥 2 − 7𝑥 + 10
It Gets Cooler
𝑓
𝑥 =
𝑥2
𝑥+1
 Graph
this function.
 Zoom
out.
 Zoom
out again.
 Keep
zooming!
Oblique/Slant Asymptotes
 We
call these oblique or slant asymptotes.
 What
 It
does oblique mean?
means slanted.
 These
occur when a graph approaches a
linear relationship at its ends.
Oblique/Slant Asymptotes
𝑓 be a rational function defined as:
𝑎 𝑥
𝑓 𝑥 =
𝑏 𝑥
 If 𝑛 = 𝑚 + 1, the graph has an oblique
asymptote.
 Let
𝑓
𝑥 =
 The
𝑎 𝑥
𝑏 𝑥
=𝑞 𝑥 +
𝑟 𝑥
𝑏 𝑥
asymptote will run along the line 𝑦 = 𝑞(𝑥).
Slant Asymptote
𝑥2
𝑥+1

𝑓 𝑥 =


We know this has a vertical asymptote at 𝑥 = −1.
Because 𝑛 > 𝑚, there are no horizontal asymptotes.
Because 𝑛 = 𝑚 + 1, there is a slant asymptote.
Do the long division:

Slant asymptote at 𝑦 = 𝑥 − 1.


Check the Graph
Determine any asymptotes, intercepts,
the domain of the function
ℎ
𝑝
𝑥 =
𝑥 2 +3𝑥−3
𝑥+4
𝑥 =
𝑥 2 −4𝑥+1
2𝑥−3
More Practise
𝑓
𝑥 =
𝑥 3 −2𝑥 2 +𝑥+18
𝑥 2 −12𝑥+35
Determine any asymptotes, intercepts,
the domain of the function
𝑔
𝑐
𝑥 =
𝑥 2 +10𝑥+24
𝑥 2 +𝑥−12
𝑥 =
𝑥 2 −2𝑥−3
𝑥 2 −4𝑥−5