MHF4U Grade 12 Advanced Functions – Rational Functions Test

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.
      • 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.


Reciprocal of a Quadratic Function

  • Rational 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.


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


Solve Rational functions 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.
  • 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