Name: 

Find a quadratic function to model the values in the table. Predict the value of y for x = 6.

x	y
–1	2
0	–2
3	10

(–2, 8), (0, –4), (4, 68)

A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lake’s population of waterfowl on each of the next six weeks.

Week	0	1	2	3	4	5	6
Population	585	582	629	726	873	1,070	1,317

a.	Find a quadratic function that models the data as a function of x, the number of weeks.
b.	Use the model to estimate the number of waterfowl at the lake on week 8.

A manufacturer determines that the number of drills it can sell is given by the formula , where p is the price of the drills in dollars.

a.	At what price will the manufacturer sell the maximum number of drills?
b.	What is the maximum number of drills that can be sold?

Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula , where x is the number of units produced per week, in thousands.

a.	How many units should the company produce per week to earn the maximum profit?
b.	Find the maximum weekly profit.

Which is the graph of ?

Use vertex form to write the equation of the parabola.

Identify the vertex and the y-intercept of the graph of the function .

Write in vertex form.

vertex (–4, 3), point (4, 131)

vertex (0, 3), point (–4, –45)

Solve by factoring.
= 0

The function models the height y in feet of a stone t seconds after it is dropped from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a second.

Use a graphing calculator to solve the equation . If necessary, round to the nearest hundredth.

A landscaper is designing a flower garden in the shape of a trapezoid. She wants the length of the shorter base to be 3 yards greater than the height, and the length of the longer base to be 5 yards greater than the height. For what height will the garden have an area of 360 square yards? Round to the nearest tenth of a yard.

Simplify using the imaginary number i.

–6 –

Find .

Identify the graph of the complex number .

Find the additive inverse of .

Find the first three output values of the fractal-generating function . Use z = 0 as the first input value.

Two complex numbers a + bi and c + di are equal when a = c and b = d. Solve the equation for x and y, where x and y are real numbers.

Find the missing value to complete the square.

The function models the daily profit a barbershop makes from haircuts that include a shampoo. Here P is the profit in dollars, and h is the price of a haircut with a shampoo. Write the function in vertex form. Use the vertex form to find the price that yields the maximum daily profit and the amount of the daily profit.

A landscaper is designing a flower garden in the shape of a trapezoid. She wants the shorter base to be 3 yards greater than the height and the longer base to be 7 yards greater than the height. She wants the area to be 155 square yards. The situation is modeled by the equation . Use the Quadratic Formula to find the height that will give the desired area. Round to the nearest hundredth of a yard.

In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again.

a.	Find a quadratic model for the data in the table.
b.	Use the model to estimate the population of bacteria at 9 hours.

Time (hours)	0	1	2	3	4	5	6
Population (1000s)	5.1	3.03	1.72	1.17	1.38	2.35	4.08

The table shows the number of copies of a book sold per 100,000 people in the United States for five selected years. The values in the first column are years since 1987, so corresponds to 1987, corresponds to 1990, and so on.

Years since 1987 (x)	Copies sold per 100,000 people (y)
0	8.3
3	9.4
6	9.5
9	7.4
12	5.7

a.	Use a graphing calculator to model the data with a quadratic function. Round the coefficients and constant term to four decimal places.
b.	Graph the data and the quadratic function.
c.	Use the graph or the equation to estimate the number of copies sold per 100,000 people in 1998.
d.	Would you use the quadratic function to predict the number of copies sold per 100,000 people in 2005? Explain.

Graph .

Graph . Identify the vertex and the axis of symmetry.

Graph . What is the minimum value of the function?

Graph . Does the function have a maximum or minimum value? What is this value?

A science museum is going to put an outdoor restaurant along one wall of the museum. The restaurant space will be rectangular. Assume the museum would prefer to maximize the area for the restaurant.

a.	Suppose there is 120 feet of fencing available for the three sides that require fencing. How long will the longest side of the restaurant be?
b.	What is the maximum area?

Graph .

In a baseball game, an outfielder throws a ball to the second baseman. The path of the ball is modeled by the equation , where y is the height of the ball in feet after the ball has traveled x feet horizontally. The second baseman catches the ball at the same height as the height at which the outfielder released it.

a.	What was the maximum height of the ball along its path? Answer to the nearest foot.
b.	How far was the second baseman from the outfielder at the time he caught the ball?
c.	How high above the ground was the ball when it left the hand of the outfielder?

Use the graph of .

a.	If you translate the parabola to the right 2 units and down 7 units, what is the equation of the new parabola in vertex form?
b.	If you translate the original parabola to the left 2 units and up 7 units, what is the equation of the new parabola in vertex form?
c.	How could you translate the new parabola in part (a) to get the new parabola in part (b)?

Suppose you cut a small square from a square of fabric as shown in the diagram. Write an expression for the remaining shaded area. Factor the expression.

For how many integer values of a can be factored? What are they?

The Sears Tower in Chicago is 1454 feet tall. The function models the height y in feet of an object t seconds after it is dropped from the top of the building.

a.	After how many seconds will the object hit the ground? Round your answer to the nearest tenth of a second.
b.	What is the height of the object 5 seconds after it is dropped from the top of the Sears Tower?

A carpenter is cutting a board to make a brace that will go at the bottom of a storage shed wall. The brace will be in the shape of a right triangle. The hypotenuse will be 41 inches long. The longer leg will be 31 inches longer than the shorter leg.

a.	Let x be the length of the shorter leg. Write a quadratic equation that models the situation.
b.	Use factoring to solve the equation you wrote in part (a). What are the solutions?
c.	What is the length of the longer leg of the brace?

Two boats leave from the same point at the same time. One boat travels due east and the other travels due north. One boat travels 6 kilometers faster than the other. After 4 hours, the boats are 67 kilometers apart.

a.	Let x be the speed of the slower boat. Write a quadratic equation that models the situation.
b.	Use a graphing calculator to solve the equation in part (a) graphically. What are the solutions, to the nearest hundredth?
c.	What are the speeds of the boats? Round your answers to the nearest hundredth.

A local health official has determined that the function models the probability that a randomly chosen individual in the community will get the flu x days after the first reported case.

a.	Write the function in vertex form.
b.	How many days after the first reported case is the risk greatest that an individual will become infected?

Determine the type and number of solutions of .

A park planner has sketched a rectangular park in the first quadrant of a coordinate grid. Two sides of the park lie on the x- and y-axes. A trapezoidal flower bed will be bounded by the line , the x-axis, and the vertical lines and , where . The area A of the trapezoid is modeled by . Assume that lengths along the axes are measured in meters. For what value of a will the trapezoid have an area of 25 square meters? Use the Quadratic Formula to find the answer.

During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 135°F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by .

a.	Tell whether the temperature of the part will ever reach or exceed 135°F. Use the discriminant of a quadratic equation to decide.
b.	If the machine is in operation for 90 minutes before being turned off, how many times will the temperature of the part be 134°F?

Maribel is going to build a rectangular pen for her two dogs. She has 180 feet of fencing. To keep the dogs separate, she plans to put fencing down the middle of the pen to split the large rectangle into two smaller rectangles. What are the dimensions and area of the largest pen area she can use to accommodate both dogs? Show and explain your work.

Show that is equal to . Then use this to explain how you know that 5 is the minimum value of the function.

Use a graphing calculator to graph the function .

a.	What does the graph let you conclude about real number solutions of ? Explain.
b.	Substitute for x in the equation . Simplify. Is the resulting equation true? Show your work.
c.	What conclusions can you state about solutions of ? Explain.

A baseball player hits a fly ball that is caught about 4 seconds later by an outfielder. The path of the ball is a parabola. The ball is at its highest point as it passes the second baseman, who is 127 feet from home plate. About how far from home plate is the outfielder at the moment he catches the ball? Explain your reasoning.

A data processing consultant charges clients by the hour. His weekly earnings E are modeled by the function , where x is his hourly rate in dollars. Can he earn $2500 in a single week? Explain.

Consider system of equations.

Suppose the two parabolas have the same axis of symmetry. How many possible solutions does the system have? Explain.