Rational Functions
1.7 and Their Properties
Practice Set 1
Problems 1 − 4, for each rational function below, simplify in factored form, identify the domain, the
zero(s), and the y-intercept.
2𝑥𝑥 2 − 2𝑥𝑥 𝑥𝑥 2 +𝑥𝑥 −12
1. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 1
2. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2 +6𝑥𝑥+8
Domain: __________________________________ Domain: __________________________________
Zero(s): __________________________________ Zero(s): __________________________________
y-intercept: ______________________________ y-intercept: ______________________________
𝑥𝑥 2 −2𝑥𝑥 −15 𝑥𝑥 2−𝑥𝑥 −6
3. ℎ 𝑥𝑥 = 𝑥𝑥 2 − 9
4. 𝑘𝑘 𝑥𝑥 = 𝑥𝑥 2 +4𝑥𝑥+3
Domain: __________________________________ Domain: __________________________________
Zero(s): __________________________________ Zero(s): __________________________________
y-intercept: ______________________________ y-intercept: ______________________________
Problems 5 − 8, Use the graph of each function to complete the arrow notation statements.
𝑥𝑥−3
5. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 −5𝑥𝑥+6 There is a hole in the graph at: _______________
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→2−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→2+
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→3−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→3+
Domain: _____________________________________
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3𝑥𝑥 2−8𝑥𝑥−3
6. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2− 9 There is a hole in the graph at: _______________
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−3−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−3+
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→3−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→3+
Domain: _____________________________________
𝑥𝑥 2 −12𝑥𝑥−28
7. ℎ 𝑥𝑥 = 𝑥𝑥 2 − 4 There is a hole in the graph at: _______________
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−2−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−2+
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→2−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→2+
Domain: _____________________________________
2𝑥𝑥 2 −8𝑥𝑥+6
8. 𝑘𝑘 𝑥𝑥 = 𝑥𝑥 2 +𝑥𝑥 −2 There is a hole in the graph at: _______________
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→∞
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−2−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→−2+
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→1−
lim 𝑓𝑓 𝑥𝑥 = ______________
𝑥𝑥→1+
Domain: _____________________________________
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Rational Functions
1.7 and Their Properties
Practice Set 2
Problems 9 – 12, Find the holes, vertical asymptote(s) and x-intercepts for each function. Sketch.
𝑥𝑥+2 2𝑥𝑥 2+7𝑥𝑥+6
9. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2+5𝑥𝑥+6 10. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2−𝑥𝑥 −6
2𝑥𝑥 2 +5𝑥𝑥+2 𝑥𝑥 2+6𝑥𝑥 −16
11. ℎ 𝑥𝑥 = 2𝑥𝑥 2−5𝑥𝑥−3 12. 𝑘𝑘 𝑥𝑥 = 𝑥𝑥 2 −4
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Problems 13 – 16, Find the horizontal asymptotes for the given functions and provide a reason.
3𝑥𝑥 2 +5𝑥𝑥−2 𝑥𝑥 2 +6𝑥𝑥+8
13. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 −5𝑥𝑥+6
14. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 3 −1
𝑥𝑥−5 2𝑥𝑥 3 + 8
15. ℎ 𝑥𝑥 = 𝑥𝑥+4 16. 𝑘𝑘 𝑥𝑥 = 4𝑥𝑥 3 − 12
Problems 17 – 20, Find the slant asymptote equation, if one exists, or give a reason if not possible.
𝑥𝑥 2 −5𝑥𝑥 −6 2𝑥𝑥 3 +5𝑥𝑥 −3
17. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 + 2
18. 𝑘𝑘 𝑥𝑥 = 𝑥𝑥 − 4
2𝑥𝑥 3 +3𝑥𝑥−5 3𝑥𝑥 2 −7𝑥𝑥+6
19. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥−3
20. ℎ 𝑥𝑥 = 𝑥𝑥 −6
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Problems 21 – 22, For the given functions, analyze and graph.
𝑥𝑥 2+𝑥𝑥−1
21. 𝑓𝑓 𝑥𝑥 = 𝑥𝑥−1
A. Vertical asymptote: ____________________
B. Horizontal asymptote: __________________
C. Slant asymptote: _______________________
D. y-intercept: _____________________
E. x-intercept(s): _________________________
F. Hole(s) in the graph: ___________________
G. Domain: ________________________________
H. Range: _________________________________
2𝑥𝑥 2 −4𝑥𝑥−6
22. 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 2−3𝑥𝑥−4
A. Vertical asymptote: ____________________
B. Horizontal asymptote: __________________
C. Slant asymptote: _______________________
D. y-intercept: _____________________
E. x-intercept(s): _________________________
F. Hole(s) in the graph: ___________________
G. Domain: ________________________________
H. Range: _________________________________
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Free Response Table Problem
𝒙𝒙 −80 −6 −4 −3 −2.005 −2 −1.995 0 1 1.995 2 2.005 6 80
𝒇𝒇(𝒙𝒙) 1.948 1 0 −2 −798 undefined 802 4 3.333 3.001 undefined 2.998 2.5 2.048
2𝑥𝑥 2 + 4𝑥𝑥 − 16
23. The table above represents values on the graph of the function 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 4
A. For what value(s) of x does the graph of 𝑓𝑓(𝑥𝑥) have a vertical asymptote? Give a reason for your
answer.
B. Does the function contain a point discontinuity? If so, name the coordinates of the hole. Explain.
C. Use limit notation to describe the end behavior of the function.
D. Consider the values in the table, what factor will be guaranteed in the numerator. Give a reason
for your answer.
E. Use the equation for the function and factor completely. Explain the connections to your
conclusions in parts A-D and your work.
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Free Response Application Problem
24. The population 𝑃𝑃(𝑡𝑡) of an endangered species of turtles, t years after being introduced into a
263
safe habitat area is given by 𝑃𝑃 𝑡𝑡 = 1+3.62𝑒𝑒 −0.144𝑡𝑡
A. Find the y-intercept of 𝑃𝑃 𝑡𝑡 . Interpret this value in the context of the problem.
B. How many years will it take for the initial population to double in size?
C. How many turtles are expected to be in the habitat after 10 years?
D. Find lim 𝑃𝑃 𝑡𝑡 and interpret your answer in the context of the problem.
𝑡𝑡→∞
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