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University of Toronto Scarborough
Ken Butler

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 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.  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  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 numerator.
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