$y={(x-1)}^2+2$

$y={(x+2)}^2-3$

$y=-2{(x-3)}^2+4$

$y=-{\displaystyle \frac{1}{2}}{(x+1)}^2+2$

$y=-2x^2+3$

$y=2x^2+4x-1$

$y=2x^2-4x+3$

$y=-2x^2-x+3$

$y=3x^2-2x+1$

$y=3x^2-5x+4$

$f(x)=3-4x+x^2$

$f(x)=8x-4x^2-3$

$f(x)=-2x^2-3x+2$

$f(x)={\displaystyle \frac{1}{2}}{x}^2-{\displaystyle \frac{3}{4}}x+2$

$f(x)={(x+1)}^3-2$

$f(x)={(x+1)}^4+2$

$f(x)={(1-x)}^3+1$

$f(x)=x^4+3$

$f(x)=x(x-1)(x+2)$

$f(x)=x^3-x$

$f(x)=-x^2{(x-1)}^2$

$f(x)=-x^3{(x-2)}^2$

$f(x)=-x^2(x^2-1)$

$f(x)=-{(x-1)}^2(x^2+1)$

${\displaystyle \frac{6x^2+5x-13}{3x-2}}$

${\displaystyle \frac{8x^2-14x+15}{2x-3}}$

${\displaystyle \frac{8x^4-4x^3+2x^2-7x+165}{x+1}}$

${\displaystyle \frac{x^3-3x^2+4x+7}{x^2-2x+6}}$

${\displaystyle \frac{x^3-12x+3}{x-3}}$

${\displaystyle \frac{-4x^3+3x^2-5x}{x-6}}$

${\displaystyle \frac{2x^4-3x^3+5x^2-7x+165}{x+1}}$

${\displaystyle \frac{3x^5-2x^4+x^2-16x-132}{x+2}}$

$f(x)=x^3-3x^2+11x-29;c=2$

$f(x)=2x^3+x^2-15x-2;c=-2$

$f(x)=x^4-2x^2-5x+10;c=-3$

$f(x)=x^5+2;c=1$

$f(x)=x^3-7x^2+14x-8;c=2$

$f(x)=2x^3-3x^2-12x+4;c=-2$

$f(x)=3x^3+14x^2+13x-6;c={\displaystyle \frac{1}{3}}$

$f(x)=4x^3+19x^2-13x+2;c={\displaystyle \frac{1}{4}}$

$f(x)=x^4+3x^3-x^2-9x-6$

$f(x)=9x^3-36x^2-4x+16$

$f(x)=5x^3+11x^2+2x$

$f(x)=x^3+2x^2-5x-6$

$f(x)=x^3+3x^2-4x-12$

$f(x)=2x^3-9x^2+12x-5$

$f(x)=x^3-4x^2-5x+14$

$f(x)=x^3-5x^2+3x+1$

$f(x)=2x^3-5x^2-2x+2$

$f(x)=3x^3-29x^2+29x+13$

$f(x)=x^3-7x+6;$ one zero is 2.

$f(x)=x^4+x^3-3x^2-x+2;$ 1 is a zero of multiplicity 2.

$f(x)=x^4-2x^3+6x^2-18x-27;$ two zeros are $-1$ and 3.

$f(x)=4x^3-19x^2+32x-15;$ one zero is $2-i.$

$f(x)=x^4+2x^3+9x^2+8x+20;$ one zero is $-1+2i.$

$f(x)=x^5-7x^4+24x^3-32x^2+64;2+2i$ is a zero of multiplicity 2.

$x^3-x^2-4x+4=0$

$2x^3+x^2-12x-6=0$

$4x^3-7x-3=0$

$x^3+x^2-8x-6=0$

$x^3-8x^2+23x-22=0$

$x^3-3x^2+8x+12=0$

$3x^3-5x^2+16x+6=0$

$2x^3-9x^2+18x-7=0$

$x^4-x^3-x^2-x-2=0$

$x^4-x^3-13x^2+x+12=0$

$2x^4-x^3-2x^2+13x-6=0$

$3x^4-14x^3+28x^2-10x-7=0$

$x^3+13x^2-6x-2=0$

$3x^4-9x^3-2x^2-15x-5=0$

$x^3+6x^2-28=0$

$x^3+3x^2-3x-7=0$

$f(x)=1+{\displaystyle \frac{1}{x}}$

$f(x)={\displaystyle \frac{2-x}{x}}$

$f(x)={\displaystyle \frac{x}{x^2-1}}$

$f(x)={\displaystyle \frac{x^2-9}{x^2-4}}$

$f(x)={\displaystyle \frac{x^3}{x^2-9}}$

$f(x)={\displaystyle \frac{x+1}{x^2-2x-8}}$

$f(x)={\displaystyle \frac{x^4}{x^2-4}}$

$f(x)={\displaystyle \frac{x^2+x-6}{x^2-x-12}}$

Variation. Assuming that y varies directly as x and $y=12$ when $x=4,$ find y when $x=5.$

Variation. Assuming that p varies inversely as q and $p=4$ when $q=3,$ find p when $q=4.$

Variation. Assuming that s varies directly as the square of t and $s=20$ when $t=2,$ find s when $t=3.$

Variation. Assuming that y varies inversely as $x^2$ and $y=3$ when $x=8,$ find x when $y=12.$

Missile path. A missile fired from the origin of a coordinate system follows a path described by the equation $y=-{\displaystyle \frac{1}{10}}{x}^2+20x,$ where x is in yards and the x-axis is on the ground. Sketch the missile’s path and determine what its maximum altitude is and where it hits the ground.

Minimizing area. Suppose a wire 20 centimeters long is to be cut into two pieces, each of which will be formed into a square. Find the size of each piece that minimizes the total area.

A farmer wants to fence off three identical adjoining rectangular pens, each 400 square feet in area. See the figure. What should the width and length of each pen be so that the least amount of fence is used?

Suppose the outer boundary of the pens in Exercise 87 requires heavy fence that costs $5 per foot and two internal partitions that each cost $3 per foot. What dimensions x and y will minimize the cost?

Electrical circuit. In the circuit shown in the figure, the voltage $V=100$ volts and the resistance $R=50$ ohms. We want to determine the size of the remaining resistor (x ohms). The power absorbed by the circuit is given by

1. Graph the function $y=p(x).$

2. Use a graphing calculator to find the value of x that maximizes the power absorbed.

Maximizing area. A sheet of paper for a poster is 18 square feet in area. The margins at the top and bottom are 9 inches each, and the margin on each side is 6 inches. What should the dimensions of the paper be if the printed area is to be a maximum?

Maximizing profit. A manufacturer makes and sells printers to retailers at $24 per unit. The total daily cost C in dollars of producing x printers is given by

1. Write the profit P as a function of x.

2. Find the number of printers the manufacturer should produce and sell to achieve maximum profit.

3. Find the average cost $\overline{C}(x)={\displaystyle \frac{C(x)}{x}}.$ Graph $y=\overline{C}(x).$

Meteorology. The function $p={\displaystyle \frac{69.1}{a+2.3}}$ relates the atmospheric pressure p in inches of mercury to the altitude a in miles from the surface of Earth.

1. Find the pressure on Mount Kilimanjaro at an altitude of 19,340 feet.

2. Is there an altitude at which the pressure is 0?

Wages. An employee’s wages are directly proportional to the time he or she has worked. Sam earned $280 for 40 hours. How much would Sam earn if he worked 35 hours?

Car’s stopping distance. The distance required for a car to come to a stop after its brakes are applied is directly proportional to the square of its speed. If the stopping distance for a car traveling 30 miles per hour is 25 feet, what is the stopping distance for a car traveling 66 miles per hour?

Illumination. The amount of illumination from a source of light varies directly as the intensity of the source and inversely as the square of the distance from the source. At what distance from a light source of intensity 300 candlepower will the illumination be one-half the illumination 6 inches from the source?

Chemistry. Charles’s Law states that at a constant pressure, the volume V of a gas is directly proportional to its temperature T (in Kelvin degrees). If a bicycle tube is filled with 1.2 cubic feet of air at a temperature of 295 K, what will the volume of the air in the tube be if the temperature rises to 310 K while the pressure stays the same?

Electrical circuits. The current I (measured in amperes) in an electrical circuit varies inversely as the resistance R (measured in ohms) when the voltage is held constant. The current in a certain circuit is 30 amperes when the resistance is 300 ohms.

1. Find the current in the circuit if the resistance is decreased to 250 ohms.

2. What resistance will yield a current of 60 amperes?

Safe load. The safe load that a circular column can support varies directly as the square root of its radius and inversely as the square of its length. A pillar with radius 4 inches and length 12 feet can safely support a 20-ton load. Find the load that a pillar of the same material with diameter 6 inches and length 10 feet can safely support.

Spread of disease. An infectious cold virus spreads in a community at a rate R (per day) that is jointly proportional to the number of people who are infected with the virus and the number of people in the community who are not infected yet. After the tenth day of the start of a certain infection, 15% of the total population of 20,000 people of Pollutville had been infected and the virus was spreading at the rate of 255 people per day.

1. Find the constant of proportionality, k.

2. At what rate is the disease spreading when one-half of the population is infected?

3. Find the number of people infected when the rate of infection reached 95 people per day.

Coulomb’s Law. The electric charge is measured in coulombs. (The charges on an electron and a proton, which are equal and opposite, are approximately $1.602\times {10}^{-19}$ coulomb.) Coulomb’s Law states that the force F between two particles is jointly proportional to their charges $q_1$ and $q_2$ and inversely proportional to the square of the distance between the two particles. Two charges are acted upon by a repulsive force of 96 units. What is the force if the distance between the particles is quadrupled?