Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that

List the intercepts of the equation (pp. 159160)

Find the real solutions of the equation (pp. 9299) 2x2 + 7x - 4 = 0. y = x2 - 9.

To complete the square of you add the number . (p. 56)

To graph you shift the graph of to the a distance of units. (pp. 244253) y

The graph of a quadratic function is called a(n) .

The vertical line passing through the vertex of a parabola is called the .

The x-coordinate of the vertex of a Z 0, is . f1x2 = ax2 + bx + c,

True or False The graph of opens up.

True or False The y-coordinate of the vertex of is f 2 .

True or False If the discriminant the graph of will touch the x-axis at its vertex.

In Problems 1118, match each graph to one the following functions. f1x2 = x2 - 1 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = -x2 - 1 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 - 2x + 1 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 + 2x + 1 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 - 2x + 2 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 + 2x CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 - 2x CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1118, match each graph to one the following functions. f1x2 = x2 + 2x + 2 CD E F GH x y 3 2 2 1 (1, 0) (1, 1) x y 2 2 2 (0, 1) x y 2 2 2 2 (1, 1) x y 3 2 2 1 (0, 1) x y 1 2 2 3 (1, 0) x y 1 3 2 (1, 1) x y 3 1 3 1 (1, 1)

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 1 4 x2

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 2x2 + 4

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 1x + 222 - 2

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 1x - 322 - 10

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = x2 + 4x + 2

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = x2 - 6x - 1

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 2x2 - 4x + 1

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 3x2 + 6x

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = -x2 - 2x

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = -2x2 + 6x + 2

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 1 2 x2 + x - 1

In Problems 1930, graph the function f by starting with the graph of and using transformations (shifting, compressing, stretching, and/or reflection). [Hint: If necessary, write f in the form ] f1x2 = 2 3 x2 + 4 3 x - 1

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 + 2x

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 - 4x

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = -x2 - 6x

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = -x2 + 4x

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 + 2x - 8

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 - 2x - 3

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 + 2x + 1

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = x2 + 6x + 9

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = 2x2 - x + 2

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = 4x2 - 2x + 1

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = -2x2 + 2x - 3

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = -3x2 + 3x - 2

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = 3x2 + 6x + 2

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = 2x2 + 5x + 3

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = -4x2 - 6x + 2

In Problems 3146, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. f1x2 = 3x2 - 8x + 2

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 4752, determine the quadratic function whose graph is given.

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = 2x2 + 12x

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = -2x2 + 12x

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = 2x2 + 12x - 3

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = 4x2 - 8x + 3

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = -x2 + 10x - 4

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = -2x2 + 8x + 3

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = -3x2 + 12x + 1

In Problems 5360, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f1x2 = 4x2 - 4x

The graph of the function has vertex at 0, 2 and passes through the point 1, 8 . Find 1 2 1 2 a, b, and c.

The graph of the function has vertex at 11, 4 and passes through the point Find 2 1-1, -82. a, b, and c.

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = 2x - 1; g1x2 = x2 - 4

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = -2x - 1; g1x2 = x2 - 9

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = -x2 + 4; g1x2 = -2x + 1

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = -x2 + 9; g1x2 = 2x + 1

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = -x2 + 5x; g1x2 = x2 + 3x - 4

In Problems 6368, for the given functions f and g, (a) Graph f and g on the same Cartesian plane. (b) Solve (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which , that is, the region below f and above g. f1x2 = -x2 + 7x - 6; g1x2 = x2 + x - 6

Answer Problems 69 and 70 using the following: A quadratic function of the form with may also be written in the form where and are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are and 1 with (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude?

Answer Problems 69 and 70 using the following: A quadratic function of the form with may also be written in the form where and are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are and 3 with (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude?

Suppose that (a) What is the vertex of f? (b) What are the x-intercepts of the graph of f? (c) Solve for x. What points are on the graph of f? (d) Use the information obtained in parts (a)(c) to graph

Suppose that (a) What is the vertex of f ? (b) What are the x-intercepts of the graph of f ? (c) Solve for x. What points are on the graph of f ? (d) Use the information obtained in parts (a)(c) to graph f1x2 = x2 + 2x - 8.

Find the point on the line that is closest to the point [Hint: Express the distance d from the point to the line as a function of x, and then find the minimum value of

Find the point on the line that is closest to the point

Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?

Maximizing Revenue The John Deere company has found that the revenue, in dollars, from sales of riding mowers is a function of the unit price p, in dollars, that it charges. If the revenue R is what unit price p should be charged to maximize revenue? What is the maximum revenue?

Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, it cost $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C (in dollars) to produce x thousand mp3 players is given by the function (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?

Minimizing Marginal Cost (See Problem 77.) The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given by (a) How many cell phones should be manufactured to minimize the marginal cost? (b) What is the minimum marginal cost?

Business The monthly revenue R achieved by selling x wristwatches is figured to be The monthly cost C of selling x wristwatches is (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

Business The daily revenue R achieved by selling x boxes of candy is figured to be The daily cost C of selling x boxes of candy is (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

Stopping Distance An accepted relationship between stopping distance, d (in feet), and the speed of a car, (in mph), is on dry, level concrete. (a) How many feet will it take a car traveling 45 mph to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you can be traveling to avoid being involved? (c) What might the term represent? Source: www2.nsta.org/Energy/fn_braking.html

Birthrate of Unmarried Women In the United States, the birthrate B of unmarried women (births per 1000 unmarried women) for women whose age is a is modeled by the function (a) What is the age of unmarried women with the highest birthrate? (b) What is the highest birthrate of unmarried women? (c) Evaluate and interpret Source: United States Statistical Abstract, 2009

. Find a quadratic function whose x-intercepts are and 2 and whose range is .

Find a quadratic function whose x-intercepts are and 5 and whose range is .

Let where a, b, and c are odd integers. If x is an integer, show that must be an odd integer. [Hint: x is either an even integer or an odd integer.]

Make up a quadratic function that opens down and has only one x-intercept. Compare yours with others in the class. What are the similarities? What are the differences?

On one set of coordinate axes, graph the family of parabolas for and Describe the characteristics of a member of this family.

On one set of coordinate axes, graph the family of parabolas for and Describe the general characteristics of this family. f1x2 = x b = -4, b = 0, b = 4. 2 + bx + 1 f1x2 = x c = -3, c = 0, c = 1. 2 + 2x + c

State the circumstances that cause the graph of a quadratic function to have no x-intercepts.

Why does the graph of a quadratic function open up if and down if

Can a quadratic function have a range of Justify your answer.

What are the possibilities for the number of times the graphs of two different quadratic functions intersect?