4.1.1: Rational Functions (Exercises)
Match each equation form with one of the graphs.
1. \(f\left(x\right)=\dfrac{x-A}{x-B}\) 2. \(g\left(x\right)=\dfrac{\left(x-A\right)^{2} }{x-B}\) 3. \(h\left(x\right)=\dfrac{x-A}{\left(x-B\right)^{2} }\) 4. \(k\left(x\right)=\dfrac{\left(x-A\right)^{2} }{\left(x-B\right)^{2} }\)
For each function, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal asymptote. Use that information to sketch a graph.
5.\(p\left(x\right)=\dfrac{2x-3}{x+4}\)
6. \(q\left(x\right)=\dfrac{x-5}{3x-1}\)
7. \(s\left(x\right)=\dfrac{4}{\left(x-2\right)^{2} }\)
8. \(r\left(x\right)=\dfrac{5}{\left(x+1\right)^{2} }\)
9. \(f\left(x\right)=\dfrac{3x^{2} -14x-5}{3x^{2} +8x-16}\)
10. \(g\left(x\right)=\dfrac{2x^{2} +7x-15}{3x^{2} -14x+15}\)
11. \(a\left(x\right)=\dfrac{x^{2} +2x-3}{x^{2} -1}\)
12. \(b\left(x\right)=\dfrac{x^{2} -x-6}{x^{2} -4}\)
13. \(h\left(x\right)=\dfrac{2x^{2} +\; x-1}{x-4}\)
14. \(k\left(x\right)=\dfrac{2x^{2} -3x-20}{x-5}\)
15. \(n\left(x\right)=\dfrac{3x^{2} +4x-4}{x^{3} -4x^{2} }\)
16. \(m\left(x\right)=\dfrac{5-x}{2x^{2} +7x+3}\)
17. \(w\left(x\right)=\dfrac{\left(x-1\right)\left(x+3\right)\left(x-5\right)}{\left(x+2\right)^{2} (x-4)}\)
18. \(z\left(x\right)=\dfrac{\left(x+2\right)^{2} \left(x-5\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}\)
Write an equation for a rational function with the given characteristics.
19. Vertical asymptotes at \(x=5\) and \(x=-5\)
\(x\) intercepts at \((2,\; 0)\) and \((-1,\; 0)\) \(y\) intercept at \(\left(0,\; 4\right)\)
20. Vertical asymptotes at \(x=-4\) and \(x=-1\)
\(x\) intercepts at \(\left(1,\; 0\right)\) and \(\left(5,\; 0\right)\) \(y\) intercept at \((0,\; 7)\)
21. Vertical asymptotes at \(x=-4\) and \(x=-5\)
\(x\) intercepts at \(\left(4,\; 0\right)\) and \(\left(-6,\; 0\right)\) Horizontal asymptote at \(y=7\)
22. Vertical asymptotes at \(x=-3\) and \(x=6\)
\(x\) intercepts at \(\left(-2,\; 0\right)\) and \(\left(1,\; 0\right)\) Horizontal asymptote at \(y=-2\)
23. Vertical asymptote at \(x=-1\)
Double zero at \(x=2\) \(y\) intercept at \((0,\; 2)\)
24. Vertical asymptote at \(x=3\)
Double zero at \(x=1\) \(y\) intercept at \((0,\; 4)\)
Write an equation for the function graphed.
25. 26.
27. 28.
Write an equation for the function graphed.
29. 30.
31. 32.
33. 34.
35. 36.
Write an equation for the function graphed.
37. 38.
Find the oblique asymptote of each function.
39. \(f(x)=\dfrac{3x^{2} +4x}{x+2}\)
40. \(g(x)=\dfrac{2x^{2} +3x-8}{x-1}\)
41. \(h(x)=\dfrac{x^{2} -x-3}{2x-6}\)
42. \(k(x)=\dfrac{5+x-2x^{2} }{2x+1}\)
43. \(m(x)=\dfrac{-2x^{3} +x^{2} -6x+7}{x^{2} +3}\)
44. \(n(x)=\dfrac{2x^{3} +x^{2} +x}{x^{2} +x+1}\)
45. A scientist has a beaker containing 20 mL of a solution containing 20% acid. To dilute this, she adds pure water.
a. Write an equation for the concentration in the beaker after adding \(n\) mL of water.
b. Find the concentration if 10 mL of water has been added.
c. How many mL of water must be added to obtain a 4% solution?
d. What is the behavior as \(n \to \infty\), and what is the physical significance of this?
46. A scientist has a beaker containing 30 mL of a solution containing 3 grams of potassium hydroxide. To this, she mixes a solution containing 8 milligrams per mL of potassium hydroxide.
a. Write an equation for the concentration in the tank after adding \(n\) mL of the second solution.
b. Find the concentration if 10 mL of the second solution has been added.
c. How many mL of water must be added to obtain a 50 mg/mL solution?
d. What is the behavior as \(n\to \infty\), and what is the physical significance of this?
47. Oscar is hunting magnetic fields with his gauss meter, a device for measuring the strength and polarity of magnetic fields. The reading on the meter will increase as Oscar gets closer to a magnet. Oscar is in a long hallway at the end of which is a room containing an extremely strong magnet. When he is far down the hallway from the room, the meter reads a level of 0.2. He then walks down the hallway and enters the room. When he has gone 6 feet into the room, the meter reads 2.3. Eight feet into the room, the meter reads 4.4. [UW]
a. Give a rational model of form \(m\left(x\right)=\dfrac{ax+b}{cx+d}\) relating the meter reading \(m(x)\) to how many feet \(x\) Oscar has gone into the room.
b. How far must he go for the meter to reach 10? 100?
c. Considering your function from part (a) and the results of part (b), how far into the room do you think the magnet is?
48. The more you study for a certain exam, the better your performance on it. If you study for 10 hours, your score will be 65%. If you study for 20 hours, your score will be 95%. You can get as close as you want to a perfect score just by studying long enough. Assume your percentage score, \(p(n)\), is a function of the number of hours, \(n\), that you study in the form \(p(n)=\dfrac{an+b}{cn+d}\). If you want a score of 80%, how long do you need to study? [UW]
49. A street light is 10 feet north of a straight bike path that runs east-west. Olav is bicycling down the path at a rate of 15 miles per hour. At noon, Olav is 33 feet west of the point on the bike path closest to the street light. (See the picture). The relationship between the intensity C of light (in candlepower) and the distance \(d\) (in feet) from the light source is given by \(C=\dfrac{k}{d^{2} }\), where \(k\) is a constant depending on the light source. [UW]
a. From 20 feet away, the street light has an intensity of 1 candle. What is \(k\)?
b. Find a function which gives the intensity of the light shining on Olav as a function of time, in seconds.
c. When will the light on Olav have maximum intensity?
d. When will the intensity of the light be 2 candles?
- Answer
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1. D
3. A
Vertical asymptotes Horizontal Asymptote Vertical \(y\)-Intercept Horizontal \(x\)- intercept 5. \(x = -4\) \(y = 2\) (0, -3/4) (3/2, 0) 7. \(x = 2\) \(y = 0\) (0, 1) DNE 9. \(x = -4, 1\dfrac{1}{3}\) \(y = 1\) (0, 5/16) (-1/3, 0), (5, 0) 11. \(x = -1\), hole at \(x = 1\) \(y = 1\) (0, 3) (-3, 0) 13. \(x = 4\) none
\(y = 2x\) (obique)(0, 1/4) (-1, 0), (1/2, 0) 15. \(x = 0, 4\) \(y = 0\) DNE (-2, 0), (2/3, 0) 17. \(y = -2, 4\) \(y = 1\) (0, -15/16) (1, 0), (-3, 0), (5, 0) 5.
7.
9.
11.
13.
15.
17.
19. \(y = \dfrac{50(x - 2)(x + 1)}{(x + 5)(x - 5)}\)
21. \(y = \dfrac{7(x - 4)(x + 6)}{(x + 4)(x + 5)}\)
23. \(y = \dfrac{1(x - 2)^2}{2(x + 1)}\)
25. \(y = \dfrac{4(x - 3)}{(x + 3)(x - 4)}\)
27. \(y = \dfrac{27(x - 2)}{(x + 3)(x - 3)^2}\)
29. \(y = \dfrac{1(x + 3)(x - 2)}{3(x - 1)}\)
31. \(y = \dfrac{-6(x - 1)^2}{(x + 3)(x - 2)^2}\)
33. \(y = -\dfrac{2(x)(x - 3)}{(x + 3)(x - 4)}\)
35. \(y = \dfrac{2(x - 1)^3}{(x + 1)(x - 2)^2}\)
37. \(y = \dfrac{(x - 4)(x - 2)}{(x - 4)(x + 1)}\)
39. \(y = 3x - 2\)
41. \(y = \dfrac{1}{2} x + 1\)
43. \(y = -2x + 1\)
45. a. \(C(n) = \dfrac{4}{20 + n}\)
b. \(C(10) \approx 13.33%\)
c. 80 mL
d. as \(n \to \infty\), \(C \to 0\)