MHF4UInvestigating Rational Functions Worksheet Solutions
1.
Characteristics / Function: g(x) = x+6 / Rational function:f(x) = _1__
x+6
Graph(use graphing technology for this) / /
Domain
(put this in set notation) / /
Range
(put this in set notation) / /
Positive interval / x>-6 / x>-6
Negative interval / x<-6 / x<-6
Interval(s) of increase / / None
Interval(s) of decrease / None /
Vertical asymptote(s)
(give the equation) / None / x=-6
Horizontal asymptote
(give the equation) / None / y=0
Quadrants (In which quadrant does the graph begin? Finish?) / Start: quadrant 3
Finish: quadrant 1 / Start: quadrant 3
Finish: quadrant 1
2.
Characteristics / Function: g(x) = x2 / Rational function:f(x) = _1__
x2
Graph(use graphing technology for this / /
Domain
(put this in set notation) / /
Range
(put this in set notation) / /
Positive interval / /
Negative interval / None / None
Interval(s) of increase / x>0 / x<0
Interval(s) of decrease / x<0 / x>0
Vertical asymptote(s)
(give the equation) / None / x=0
Horizontal asymptote
(give the equation) / None / y=0
Quadrants (In which quadrant does the graph begin? Finish?) / Start: quadrant 2
Finish: quadrant 1 / Start: quadrant 2
Finish: quadrant 1
3.
Characteristics / Function: g(x) = x2-16 / Rational function:f(x) = _1__
x2-16
Graph(use graphing technology for this / /
Domain
(put this in set notation) / /
Range
(put this in set notation) / /
Positive interval / /
Negative interval / /
Interval(s) of increase / /
Interval(s) of decrease / /
Vertical asymptote(s)
(give the equation) / None / x=-4
x=4
Horizontal asymptote
(give the equation) / None / y=0
Quadrants(In which quadrant does the graph begin? Finish?) / Start: quadrant 2
Finish: quadrant 1 / Start: quadrant 2
Finish: quadrant 1
4. Making connections:
Look at each pair of functions in your charts.
- Do the quadrants of g(x) and f(x) relate? If so, how? How would knowing the quadrants of a polynomial function allow you to graph the rational function? Test your theory by creating more examples using Graphcalc.Is your reasoning valid?
Answer: The polynomial and the rational functions cross through identical quadrants. If you know the quadrants for the polynomial function, you then know the quadrants for its corresponding rational function (the one formed by taking the reciprocal of the polynomial function).
- Look at the other characteristics.Are there any other relationships between g(x) and f(x)?
Answer: Positive and negative intervals are identical for both functions.
The functions have opposite intervals of increase and decrease. That is, when the polynomial function is increasing, the rational function is decreasing, if it exists in the interval.
- How can you find the vertical asymptote without graphing the function? Does this work for every rational function?
Answer: By setting the denominator to zero in the Rational Function, we can find the vertical asymptote. This is valid for all rational functions, because the function is undefined for all x values making the denominator zero. Note that the zeroes of the corresponding polynomial function are the locations of vertical asymptotes in the Rational Function.
- What can you say about the horizontal asymptote for a rational function? Is this true for every rational function?
Answer: The rational functions that we have investigated thus far with "1" in the numerator and either a linear or quadratic denominator will have a horizontal asymptote of y=0.
Page 1 of 4