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$f\left(x\right)=\frac{x}{2x+1}$
Local behavior: $\text{\hspace{0.17em}}x\to -{\frac{1}{2}}^{+},f(x)\to -\infty ,x\to -{\frac{1}{2}}^{-},f(x)\to \infty \text{\hspace{0.17em}}$
End behavior: $\text{\hspace{0.17em}}x\to \pm \infty ,f(x)\to \frac{1}{2}$
$f\left(x\right)=\frac{2x}{x-6}$
$f\left(x\right)=\frac{-2x}{x-6}$
Local behavior: $\text{\hspace{0.17em}}x\to {6}^{+},f(x)\to -\infty ,x\to {6}^{-},f(x)\to \infty ,\text{\hspace{0.17em}}$ End behavior: $\text{\hspace{0.17em}}x\to \pm \infty ,f(x)\to -2$
$f\left(x\right)=\frac{{x}^{2}-4x+3}{{x}^{2}-4x-5}$
$f\left(x\right)=\frac{2{x}^{2}-32}{6{x}^{2}+13x-5}$
Local behavior: $\text{\hspace{0.17em}}x\to -{\frac{1}{3}}^{+},f(x)\to \infty ,x\to -{\frac{1}{3}}^{-},\text{\hspace{0.17em}}$ $f(x)\to -\infty ,x\to {\frac{5}{2}}^{-},f(x)\to \infty ,x\to {\frac{5}{2}}^{+}$ , $f(x)\to -\infty $
End behavior:
$x\to \pm \infty ,$
$f(x)\to \frac{1}{3}$
For the following exercises, find the slant asymptote of the functions.
$f(x)=\frac{24{x}^{2}+6x}{2x+1}$
$f(x)=\frac{4{x}^{2}-10}{2x-4}$
$y=2x+4$
$f(x)=\frac{81{x}^{2}-18}{3x-2}$
$f(x)=\frac{6{x}^{3}-5x}{3{x}^{2}+4}$
$y=2x$
$f(x)=\frac{{x}^{2}+5x+4}{x-1}$
For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.
The reciprocal function shifted up two units.
$V.A.\text{}x=0,H.A.\text{}y=2$
The reciprocal function shifted down one unit and left three units.
The reciprocal squared function shifted to the right 2 units.
$V.A.\text{}x=2,\text{}H.A.\text{}y=0$
The reciprocal squared function shifted down 2 units and right 1 unit.
For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.
$p\left(x\right)=\frac{2x-3}{x+4}$
$V.A.\text{}x=-4,\text{}H.A.\text{}y=2;\left(\frac{3}{2},0\right);\left(0,-\frac{3}{4}\right)$
$q\left(x\right)=\frac{x-5}{3x-1}$
$s\left(x\right)=\frac{4}{{\left(x-2\right)}^{2}}$
$V.A.\text{}x=2,\text{}H.A.\text{}y=0,\text{}(0,1)$
$r\left(x\right)=\frac{5}{{\left(x+1\right)}^{2}}$
$f\left(x\right)=\frac{3{x}^{2}-14x-5}{3{x}^{2}+8x-16}$
$V.A.\text{}x=-4,\text{}x=\frac{4}{3},\text{}H.A.\text{}y=1;(5,0);\left(-\frac{1}{3},0\right);\left(0,\frac{5}{16}\right)$
$g\left(x\right)=\frac{2{x}^{2}+7x-15}{3{x}^{2}-14+15}$
$a\left(x\right)=\frac{{x}^{2}+2x-3}{{x}^{2}-1}$
$V.A.\text{}x=-1,\text{}H.A.\text{}y=1;\left(-3,0\right);\left(0,3\right)$
$b\left(x\right)=\frac{{x}^{2}-x-6}{{x}^{2}-4}$
$h\left(x\right)=\frac{2{x}^{2}+x-1}{x-4}$
$V.A.\text{}x=4,\text{}S.A.\text{}y=2x+9;\left(-1,0\right);\left(\frac{1}{2},0\right);\left(0,\frac{1}{4}\right)$
$k\left(x\right)=\frac{2{x}^{2}-3x-20}{x-5}$
$w\left(x\right)=\frac{\left(x-1\right)\left(x+3\right)\left(x-5\right)}{{\left(x+2\right)}^{2}(x-4)}$
$V.A.\text{}x=-2,\text{}x=4,\text{}H.A.\text{}y=1,\left(1,0\right);\left(5,0\right);\left(-3,0\right);\left(0,-\frac{15}{16}\right)$
$z\left(x\right)=\frac{{\left(x+2\right)}^{2}\left(x-5\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$
For the following exercises, write an equation for a rational function with the given characteristics.
Vertical asymptotes at $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-5,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}(2,0)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(-1,0),\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,4\right)$
$y=50\frac{{x}^{2}-x-2}{{x}^{2}-25}$
Vertical asymptotes at $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ x- intercepts at $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,0\right),\text{\hspace{0.17em}}$ y- intercept at $\text{\hspace{0.17em}}(0,7)$
Vertical asymptotes at $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-5,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}\left(4,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-6,0\right),\text{\hspace{0.17em}}$ Horizontal asymptote at $\text{\hspace{0.17em}}y=7$
$y=7\frac{{x}^{2}+2x-24}{{x}^{2}+9x+20}$
Vertical asymptotes at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=6,\text{\hspace{0.17em}}$ x -intercepts at $\text{\hspace{0.17em}}\left(-2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,0\right),\text{\hspace{0.17em}}$ Horizontal asymptote at $\text{\hspace{0.17em}}y=-2$
Vertical asymptote at $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ Double zero at $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}(0,2)$
$y=\frac{1}{2}\frac{{x}^{2}-4x+4}{x+1}$
Vertical asymptote at $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ Double zero at $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}(0,4)$
For the following exercises, use the graphs to write an equation for the function.
$y=4\frac{x-3}{{x}^{2}-x-12}$
$y=-9\frac{x-2}{{x}^{2}-9}$
$y=\frac{1}{3}\frac{{x}^{2}+x-6}{x-1}$
$y=-6\frac{{(x-1)}^{2}}{(x+3){(x-2)}^{2}}$
For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote
$f(x)=\frac{1}{x-2}$
$x$ | 2.01 | 2.001 | 2.0001 | 1.99 | 1.999 |
$y$ | 100 | 1,000 | 10,000 | –100 | –1,000 |
$x$ | 10 | 100 | 1,000 | 10,000 | 100,000 |
---|---|---|---|---|---|
$y$ | .125 | .0102 | .001 | .0001 | .00001 |
Vertical asymptote $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=0$
$f(x)=\frac{x}{x-3}$
$f(x)=\frac{2x}{x+4}$
$x$ | –4.1 | –4.01 | –4.001 | –3.99 | –3.999 |
$y$ | 82 | 802 | 8,002 | –798 | –7998 |
$x$ | 10 | 100 | 1,000 | 10,000 | 100,000 |
$y$ | 1.4286 | 1.9331 | 1.992 | 1.9992 | 1.999992 |
Vertical asymptote $\text{\hspace{0.17em}}x=-4,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=2$
$f(x)=\frac{2x}{{(x-3)}^{2}}$
$f(x)=\frac{{x}^{2}}{{x}^{2}+2x+1}$
$x$ | –.9 | –.99 | –.999 | –1.1 | –1.01 |
$y$ | 81 | 9,801 | 998,001 | 121 | 10,201 |
$x$ | 10 | 100 | 1,000 | 10,000 | 100,000 |
$y$ | .82645 | .9803 | .998 | .9998 |
Vertical asymptote $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ Horizontal asymptote $\text{\hspace{0.17em}}y=1$
For the following exercises, use a calculator to graph $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ Use the graph to solve $\text{\hspace{0.17em}}f\left(x\right)>0.$
$f(x)=\frac{2}{x+1}$
$f(x)=\frac{4}{2x-3}$
$\left(\frac{3}{2},\infty \right)$
$f(x)=\frac{2}{\left(x-1\right)\left(x+2\right)}$
$f(x)=\frac{x+2}{\left(x-1\right)\left(x-4\right)}$
$(-2,1)\cup (4,\infty )$
$f(x)=\frac{{(x+3)}^{2}}{{\left(x-1\right)}^{2}\left(x+1\right)}$
For the following exercises, identify the removable discontinuity.
$f(x)=\frac{{x}^{2}-4}{x-2}$
$\left(2,4\right)$
$f(x)=\frac{{x}^{3}+1}{x+1}$
$f(x)=\frac{{x}^{2}+x-6}{x-2}$
$\left(2,5\right)$
$f(x)=\frac{2{x}^{2}+5x-3}{x+3}$
$f(x)=\frac{{x}^{3}+{x}^{2}}{x+1}$
$\left(\u20131,\text{1}\right)$
For the following exercises, express a rational function that describes the situation.
A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes.
A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes.
$C(t)=\frac{8+2t}{300+20t}$
For the following exercises, use the given rational function to answer the question.
The concentration $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a drug in a patient’s bloodstream $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ hours after injection in given by $\text{\hspace{0.17em}}C(t)=\frac{2t}{3+{t}^{2}}.\text{\hspace{0.17em}}$ What happens to the concentration of the drug as $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ increases?
The concentration $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a drug in a patient’s bloodstream _{ $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ } hours after injection is given by $\text{\hspace{0.17em}}C(t)=\frac{100t}{2{t}^{2}+75}.\text{\hspace{0.17em}}$ Use a calculator to approximate the time when the concentration is highest.
After about 6.12 hours.
For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.
An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = length of the side of the base.
A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = length of the side of the base.
$A(x)=50{x}^{2}+\frac{800}{x}.\text{\hspace{0.17em}}$ 2 by 2 by 5 feet.
A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.
A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.
$A(x)=\pi {x}^{2}+\frac{100}{x}.\text{\hspace{0.17em}}$ Radius = 2.52 meters.
A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ = radius.
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