Asymptotes of Rational
Functions
1/21/2016
Vocab
• Continuous graph – a graph that has no breaks, jumps, or holes
• Discontinuous graph – a graph that contains breaks, jumps or holes
• Point of discontinuity – is the x-coordinate of a point where the graph
of a function is not continuous
• Non-removable discontinuity – a break in the graph of a function
where you cannot redefine the function to make the graph
continuous
• Removable discontinuity – a point of discontinuity, a, of a function
that you can remove be redefining the function at x=a
Horizontal Asymptote
• Rational functions in the form f x =
𝑃(𝑥)
𝑄(𝑥)
• To find the H.A. of f(x) you must compare the highest degree of the
numerator and denominator
• If the degree of P(x) is greater than the degree of Q(x) there is no H.A.
• If the degree of P(x) is less than the degree of Q(x) there is a H.A. at
𝑎
y = where ‘a’ is the coefficient of P(x) and ‘b’ is the coefficient of
𝑏
Q(x)
• If the degree of P(x) is equal to the degree of Q(x) there is a H.A. at
y=0
Vertical Asymptote
• Rational functions in the form f x =
𝑃(𝑥)
𝑄(𝑥)
• f(x) has vertical asymptotes at each zero of Q(x)
• Set the denominator equal to zero
• Must simplify rational expression before finding any V.A.
Holes
• Rational functions in the form f x =
𝑃(𝑥)
𝑄(𝑥)
• Holes come from the factors of P(x) and Q(x) that cancel out when
simplifying
• The factors that cancel must be set equal to zero to find the values of
‘x’ where the holes are located.
X- and Y- intercepts
• To find the y-intercept of a rational function all the ‘x’ values must be
equal to zero and solve.
• To find the x-intercepts of a rational function the numerator of the
rational function must be set equal to zero and solve for the ‘x’ values
• MUST SIMPLIFY THE RATIONAL EQUATION FIRST
Steps to finding asymptotes, holes, and
intercepts
1. Horizontal Asymptote
• Compare degree of P(x)
and Q(x)
2. Simplify
3. Find holes
• Factors that canceled
4. x- and y- intercepts
• y- intercept set all x’s equal to
zero
• x- intercept set numerator
equal to zero
5. Vertical asymptote
• Set denominator equal to
zero
EX 1 Find the Holes
•𝑦 =
𝑥 2 −3𝑥−4
𝑥−4
Ex 2 Find the V.A
•𝑦 =
𝑥+1
𝑥 2 −5𝑥+6
•𝑦 =
𝑥+3
𝑥 2 −𝑥−12
Ex 3 Find the H.A.
1. 𝑦 =
2𝑥
𝑥+3
2. 𝑦 =
𝑥−2
𝑥 2 −2𝑥−3
3. 𝑦 =
𝑥2
2𝑥−5