Unit 3: Rational Equations

3.1 Reciprocal of a Linear Function

• 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.
• 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.

 3.2 Reciprocal of a Quadratic Function

• 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 asymptotes.
• 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.

 3.3 Rational Functions of the Form  f(x) = (ax + b) / (cx + d) 

• 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 left
• 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

 3.4 Solve Rational Equations and Inequalities

• 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.

3.5 Making Connections With Rational Functions and Equations 

• 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 numerator.
• Since the equation is already factored, the roots are clearly shown. Graph or use Interval Table method to find the intervals x which satisfy the equation