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Section: 5.4
Objective(s): Graph rational functions
: Transform rational functions by changing parameters
Rational function
A function that can be written as the quotient of two polynomials with
integral coefficients, plus a constant term. The graph of a rational function is
𝑝(𝑥)
usually a hyperbola. 𝑓(𝑥) = 𝑞(𝑥) + 𝐶, where 𝑝(𝑥) and 𝑞(𝑥) are polynomials
and C is a constant, is the generic form of a rational function.
Continuous function
A function which contains no discontinuities. In mathematical terms, a
function is continuous when ∀𝑎 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛, lim 𝑓(𝑥) = 𝑓(𝑎).
𝑥→𝑎
Discontinuity
Parent function of the family of
rational functions.
A point for which a function is undefined, yet the same function is defined as
it approaches the discontinuity from either side. Discontinuities may be
vertical asymptotes or “holes”.
𝑓(𝑥) =
1
𝑥
The y-axis is a vertical asymptote. The x-axis is a horizontal asymptote.
The general graphing form of
the function:
𝑓(𝑥) =
𝑎
+ 𝑘;
𝑥−ℎ
a is the vertical stretch or compression
a < 0 is a reflection over the y-axis
h is the horizontal translation
k is the vertical translation
Describe the transformation
from the parent graph of the
following:
𝑔(𝑥) =
1
𝑥+2
The graph is translated left 2 units.
𝑓(𝑥) =
1
−3
𝑥
The graph is translated 3 units down.
ℎ(𝑥) =
5
+6
𝑥−4
The graph is stretched by a factor of 5.
The graph is translated right 4 units.
The graph is translated up 6 units.
Identifying asymptotes of
rational functions.
Vertical asymptotes
Vertical asymptotes occur at the x-values which cause the value of the
denominator to be 0. Since division by 0 is undefined, the function is
undefined at these x-values.
𝑝(𝑥)
If 𝑓(𝑥) = 𝑞(𝑥) + 𝐶 , then the vertical asymptotes exist at the x-values where
𝑞(𝑥) = 0.
Horizontal asymptotes
𝑝(𝑥)
If 𝑓(𝑥) = 𝑞(𝑥) + 𝐶, consider the largest exponent to appear in 𝑝(𝑥) and the
largest exponent to appear in 𝑞(𝑥).
If the largest exponent in 𝑝(𝑥) is higher than the largest exponent in 𝑞(𝑥),
there are no horizontal asymptotes.
If the largest exponent in 𝑝(𝑥) is higher than the largest exponent in 𝑞(𝑥),
there is a horizontal asymptote at 𝑥 = 𝐶.
If the largest exponent in the denominator is equal to the largest exponent in
the numerator, then a horizontal asymptote exists at the following ratio:
𝑥=
𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑤𝑖𝑡ℎ ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑝(𝑥)
+𝐶
𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑤𝑖𝑡ℎ ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑛 𝑞(𝑥)
Identify the vertical and
horizontal asymptotes:
𝑓(𝑥) =
3𝑥 3 + 4
−2𝑥 2
Identify vertical asymptotes:
Determine what values of x make the function undefined:
−2𝑥 2 = 0
𝑥=0
∴ a vertical asymptote exists at 𝑥 = 0.
Identify horizontal asymptotes:
Since the highest exponent in the numerator is greater than the highest
exponent in the denominator (the value of C is irrelevant), no horizontal
asymptotes exist.
𝑓(𝑥) =
5𝑥 2
2𝑥 2 − 5𝑥 + 2
Identify vertical asymptotes:
Determine what values of x make the function undefined.
2𝑥 2 − 5𝑥 + 2 = 0
(2𝑥 − 1)(𝑥 − 2) = 0
2𝑥 − 1 = 0 𝑜𝑟 𝑥 − 2 = 0
1
𝑥 = 𝑜𝑟 𝑥 = 2
2
1
∴ vertical asymptotes exist at 𝑥 = 2 and 𝑥 = 2
Identify horizontal asymptotes:
Since the highest exponent in the numerator equals the highest power in the
denominator and C = 0, a horizontal asymptote exists at the ratio of the
coefficients of those terms.
5
∴ a horizontal asymptote exists at 𝑦 = 2
𝑓(𝑥) =
𝑥+2
𝑥 2 − 3𝑥
Identify vertical asymptotes:
Determine what values of x make the function undefined.
𝑥 2 − 3𝑥 = 0
𝑥(𝑥 − 3) = 0
𝑥 = 0 𝑜𝑟 𝑥 − 3 = 0
𝑥 = 0 𝑜𝑟 𝑥 = 3
∴ vertical asymptotes exist at 𝑥 = 0 and 𝑥 = 3
Identify the horizontal asymptotes:
Since the highest exponent in the numerator is less than the highest exponent
in the denominator and C = 0, a horizontal asymptote exists at 𝑦 = 0.
Domain
The domain of a rational function is the set of all real numbers except the xvalues of the discontinuities (asymptotes or holes).
x-intercepts (roots)(zeros)
If 𝑓(𝑥) = 𝑞(𝑥) , the x-intercepts (also called the “roots” or “zeros” of the
𝑝(𝑥)
function) exist at the x-values where 𝑝(𝑥) = 0.
If 𝑓(𝑥) is a rational function, the y-intercepts are identified at 𝑓(0).
y-intercepts
“holes”
0.
Identify the discontinuities,
vertical asymptotes, horizontal
asymptotes, zeros, holes, yintercepts and domain, for the
following:
𝑓(𝑥) =
𝑝(𝑥)
If 𝑓(𝑥) = 𝑞(𝑥) , a “hole” exists at x-values where both 𝑝(𝑥) = 0 and 𝑞(𝑥) =
2
𝑥−3
Discontinuities: 3
Vertical asymptote: 𝑥 = 3
Horizontal asymptote: 𝑦 = 0
Zeros: 𝑁𝑜𝑛𝑒
Holes: None
2
y-intercepts: (0, − 3)
Domain: 𝑅 ≠ 3
𝑓(𝑥) =
𝑥 2 − 7𝑥 + 12
𝑥2 − 4
Discontinuities: 2, -2
Vertical asymptote: 𝑥 = ±2
Horizontal asymptote: 𝑦 = 1
Zeros: {3,4} or (3,0) and (4,0) (note the different notations)
y-intercepts: (0, −3)
Domain: 𝑅 ≠ ±2
𝑓(𝑥) =
𝑥 3 + 𝑥 2 − 12𝑥
+4
𝑥 3 − 9𝑥
Vertical asymptote: 𝑥(𝑥 − 3)(𝑥 + 3) = 0
𝑥 = −3 (not 3 or 0 though because they also make the
numerator 0 ∴ 𝑥 = 3 𝑎𝑛𝑑 𝑥 = 0 are holes, not
asymptotes)
Holes: 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 3
1
Horizontal asymptote: 𝑦 = 1 + 4; 𝑦 = 5
Zeros: Zeros get really messy here because the C value is not 0 (it’s 4). So
it’s not as simple as just setting the numerator equal to 0, because that will
make the y-value 4, not 0. We’ll save finding zeros on something where C
isn’t zero until pre-calc.
y-intercepts: None (because 𝑥 = 0 is a hole)
Domain: 𝑅 ≠ ±3,0