Introduction to Rational Equations
15 November 2010
Rational EquationsDefinition
anxn + an-1xn-1 + … + a2x2 + a1n1 + a0
bnxn + bn-1xn-1 + … + b2x2 + b1n1 + b0
A polynomial divided bya polynomial
Rational equations are fractions
Polynomial
Polynomial
Polynomials vs. Rationals
Polynomial Equations
Rational Equations
ContinuousSmooth
DiscontinuousHas jumps, breaks, sharp bends, and/or holes
Discontinuities
Affect the shape, domain and range of an equation
Three major types for rational equations: Vertical Asymptotes Horizontal Asymptotes Removable Discontinuities (aka
Holes)
Asymptotes
A line that the equation approaches but can never reach
Not part of the equation
Represented by a dashed line
Vertical Asymptotes
Occur when the denominator equals zero
Can never be crossed Always in the form x = Abbreviated VA
Vertical Asymptotes, cont.Hand Drawn Calculator Drawn
Horizontal Asymptotes
Occurs when the degree of the numerator ≤ the degree of the denominator Ex.
Can be crossed Always in the form y = Abbreviated HA
3x
xy
Horizontal Asymptotes, cont.
Hand Drawn Calculator Drawn
Removable Discontinuities
aka Holes Gaps in the graph at a single point Occur when Always in the form x = Represented by an open circle (or hole) in
the graph
0
0y
Removable Discontinuities, cont.
Hand Drawn Calculator Drawn
Graphing Calculators and Removable Discontinuities Graphing calculators have difficulty
showing removable discontinuities Check the table for errors!
Example
Roots:
y-int:
VA:
HA:
Holes:
2
1
x
y
Your Turn:
On the “Identifying Features of Rational Equations” handout, answer problems 1 – 8.
Don’t answer the domain and range questions!
Discontinuities and Domain and Range
Discontinuities affect the domain and range of a rational equation
Vertical Asymptotes → Domain Horizontal Asymptotes → Range Removable Discontinuities →
Domain and Range
Example 1:
Domain:
Range:
3
2
x
xy
Example 2:
Domain:
Range:
2
42
x
xy
Your Turn:
Answer the domain and range questions for problems 1 – 8 on the “Identifying Features of Rational Equations” handout.
Homework
Complete problems 1 – 6 on the “Identifying the Features of Rational Equations Practice” handout.
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