In Exercises 17 38, use the vertex and intercepts to

In Exercises 1 4, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 1 4, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 1 4, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 1 4, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 5 8, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 5 8, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 5 8, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 5 8, the graph of a quadratic function is given. Write the function s equation, selecting from the following options.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 9 16, find the coordinates of the vertex for the parabola defined by the given quadratic function.

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 17 38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

In Exercises 39 44, an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function s domain and its range.

The vertex is and the parabola opens up

The vertex is and the parabola opens down

Maximum = -6 x = 10

Minimum = 18 x = -6

In Exercises 49 52, write an equation in standard form of the parabola that has the same shape as the graph of but with the given point as the vertex

In Exercises 49 52, write an equation in standard form of the parabola that has the same shape as the graph of but with the given point as the vertex

In Exercises 49 52, write an equation in standard form of the parabola that has the same shape as the graph of but with the given point as the vertex

In Exercises 49 52, write an equation in standard form of the parabola that has the same shape as the graph of but with the given point as the vertex

In Exercises 53 56, write an equation in standard form of the parabola that has the same shape as the graph of or but with the given maximum or minimum.

In Exercises 53 56, write an equation in standard form of the parabola that has the same shape as the graph of or but with the given maximum or minimum.

In Exercises 53 56, write an equation in standard form of the parabola that has the same shape as the graph of or but with the given maximum or minimum.

In Exercises 53 56, write an equation in standard form of the parabola that has the same shape as the graph of or but with the given maximum or minimum.

When the shot whose path is shown by the blue graph is released at an angle of 35, its height, in feet, can be modeled by where is the shot s horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the blue graph. a. What is the maximum height of the shot and how far from its point of release does this occur? b. What is the shot s maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

When the shot whose path is shown by the red graph is released at an angle of 65, its height, in feet, can be modeled by where is the shot s horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph. a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur? b. What is the shot s maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

The graph at the top of the next column shows U.S. adult wine consumption, in gallons per person, for selected years from 1980 through 2005. The function f1x2 = 0.004x2 - 0.094x + 2.6

The graph shows the number of movie tickets sold in the United States, in billions, from 2000 through 2005. f1x2, x 2005 2.8 2000 2.5 1995 2.1 1990 2.0 1985 2.3 2.8 2.6 2.7 2.5 2.4 2.3 2.2 2.1 Wine Consumption (gallons per person) Wine Consumption per United States Adult Year 1980 2.6 2.0 Source: Adams Business Media 1.60 1.50 1.40 1.65 1.55 1.45 1.35 1.30 United States Movie Attendance Year 2000 1.45 2001 1.50 2002 1.63 2003 1.60 2004 1.53 2005 1.41 Movie Tickets Sold (billions) Source: National Association of Theater Owners The function models U.S. movie attendance, in billions of tickets sold, years after 2000. a. According to this function, how many billions of movie tickets were sold in 2005? Does this overestimate or underestimate the number shown by the graph? By how much? b. According to this function, in which year was movie attendance at a maximum? Round to the nearest year. What does the function give for the billions of tickets sold for that year? By how much does this differ from the number shown by the graph?

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Among all pairs of numbers whose sum is 20, find a pair whose product is as large as possible. What is the maximum product?

Among all pairs of numbers whose difference is 16, find a pair whose product is as small as possible. What is the minimum product?

Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

A rain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

On a certain route, an airline carries 8000 passengers per month, each paying $50. A market survey indicates that for each $1 increase in the ticket price, the airline will lose 100 passengers. Find the ticket price that will maximize the airline s monthly revenue for the route. What is the maximum monthly revenue?

A car rental agency can rent every one of its 200 cars at $30 per day. For each $1 increase in rate, five fewer cars are rented. Find the rental amount that will maximize the agency s daily revenue. What is the maximum daily revenue?

The annual yield per walnut tree is fairly constant at 60 pounds per tree when the number of trees per acre is 20 or fewer. For each additional tree over 20, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding. How many walnut trees should be planted per acre to maximize the annual yield for the acre? What is the maximum number of pounds of walnuts per acre?

The annual yield per cherry tree is fairly constant at 50 pounds per tree when the number of trees per acre is 30 or fewer. For each additional tree over 30, the annual yield per tree for all trees on the acre decreases by 1 pound due to overcrowding. How many cherry trees should be planted per acre to maximize the annual yield for the acre? What is the maximum number of pounds of cherries per acre?

What is a quadratic function?

What is a parabola? Describe its shape.

Explain how to decide whether a parabola opens upward or downward

Describe how to find a parabola s vertex if its equation is expressed in standard form. Give an example.

Describe how to find a parabola s vertex if its equation is in the form Use as an example

A parabola that opens upward has its vertex at (1, 2). Describe as much as you can about the parabola based on this information. Include in your discussion the number of (if any) for the parabola.

Use a graphing utility to verify any five of your hand-drawn graphs in Exercises 17 38.

a. Use a graphing utility to graph in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is Because the leading coefficient, 2, of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at the setting for should extend past this, so try and The setting for should include (and probably go below) the of the graph s minimum so try Experiment with Ymax until your utility shows the parabola s major features.

In Exercises 85 88, find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.

In Exercises 85 88, find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.

In Exercises 85 88, find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.

In Exercises 85 88, find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.

The following data show fuel efficiency, in miles per gallon, for all U.S. automobiles in the indicated year.

I must have made an error when graphing this parabola because its axis of symmetry is the

I like to think of a parabola s vertex as the point where it intersects its axis of symmetry

I threw a baseball vertically upward and its path was a parabola.

Figure 2.8 on page 293 shows that a linear function provides a better description of the football s path than a quadratic function

In Exercises 94 97, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

In Exercises 94 97, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

In Exercises 94 97, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

In Exercises 94 97, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

In Exercises 98 99, find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose is the same as the given point.

In Exercises 98 99, find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose is the same as the given point.

In Exercises 100 101, write the equation of each parabola in standard form.

In Exercises 100 101, write the equation of each parabola in standard form.

Find the point on the line whose equation is that is closest to the origin. Hint: Minimize the distance function by minimizing the expression under the square root.

A 300-room hotel can rent every one of its rooms at $80 per room. For each $1 increase in rent, three fewer rooms are rented. Each rented room costs the hotel $10 to service per day. How much should the hotel charge for each room to maximize its daily profit? What is the maximum daily profit?

A track and field area is to be constructed in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is to be 440 yards. Find the dimensions of the rectangle that maximize the area of the rectangular portion of the field.

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that initially increase and then decrease, or vice versa, and therefore can be modeled by a quadratic function. Group members should select the two sets of data that are most interesting and relevant. For each data set selected, a. Use the quadratic regression feature of a graphing utility to find the quadratic function that best fits the data. b. Use the equation of the quadratic function to make a prediction from the data. What circumstances might affect the accuracy of your prediction? c. Use the equation of the quadratic function to write and solve a problem involving maximizing or minimizing the function.

Exercises 106 108 will help you prepare for the material covered in the next section.

Exercises 106 108 will help you prepare for the material covered in the next section.

Exercises 106 108 will help you prepare for the material covered in the next section.