Unit 3: Rational Equations
The reciprocal of a linear function has the form:
f(x) = 1 / kx – c
The restriction on a domain of a reciprocal linear function can be determined by finding the
value of x that makes the denominator equal to zero, that is x = c / k.
The Vertical Asymptote of a reciprocal linear function has an equation of the form x = k / c.
The horizontal asymptote of a reciprocal linear function has equation y = 0.
If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope,
and the right branch has a negative, increasing slope.
Basically occupies Q3 and Q1.
If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow,
and the right branch has a positive, decreasing slope.
Basically occupies Q2 and Q4.
Rational quadratic functions can be analyzed using key
features:asymptotes, intercepts, slope (positive or negative, increasing or
decreasing), domain, range, and positive and negative intervals.
Reciprocal of quadratic functions with two zeros have three parts, with the middle one
reaching a maximum or minimum points. This point is equidistant from the two vertical
The behavior near asymptotes is similar to that of reciprocals of linear functions.
All of the behaviors listed above can be predicted by analyzing the roots of the quadratic
relation to the denominator.
A rational function of the form f(x) = (ax + b) / (cx + d) has the following key features:
The vertical asymptote can be found by setting the denominator equal to zero and
solving for x, provided the numerator does not have the same zero.
The horizontal asymptote can be found by dividing each term in both the numerator
and the denominator by x and investigating the behavior of the function as x -> positive
or negative infinity.
The coefficient b acts to stretch the curve but has no effect on the asymptotes, domain,
or range. The coefficient d shifts the vertical asymptote.
The two branches of the graph of the function are equidistant from the point of
intersection of the vertical and horizontal asymptotes.
Analysis of End Behavior
For vertical asymptote
Substitute a number very close to the VA from the right, and a number from the
Analyze the result of that number and express the end behavior
Whether As x -> VA +/- , y -> +/- infinity
For horizontal asymptote
Substitute a very large negative and positive number for x and analyze the
behavior of y.
Express the end behavior with the results from that substitution
As x -> +/- Infinity, y -> HA from above/below
To solve rational equations algebraically, start by factoring the expressions in the numerator
and denominator to find asymptotes and restrictions.
Next, multiply both sides by the factored denominators, and simplify to obtain a polynomial
equation. Then solve.
For Rational inequalities
Set the right side of the equation zero.
Factor the expression to find restrictions
Based on the assumption that x = a / b is true if and only if a * b = x.
On the left side of the equation, take the denominator and multiply it by the