Fill in the blanks.

Linear, constant, and squaring functions are examples of __________ functions.

J H.

Numerade Educator

Fill in the blanks.

A polynomial function of degree and leading coefficient $ a_n $ is a function of the form $ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 (a_n \neq 0) $ where $ n $ is a ________ _________ and $ a_n, a_{n-1}, \cdots , a_1, a_0 $ are ________ numbers.

J H.

Numerade Educator

Fill in the blanks.

A __________ function is a second-degree polynomial function, and its graph is called a __________.

J H.

Numerade Educator

Fill in the blanks.

The graph of a quadratic function is symmetric about its ________.

Suzanne W.

Numerade Educator

Fill in the blanks.

If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________.

J H.

Numerade Educator

Fill in the blanks.

If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________.

J H.

Numerade Educator

In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

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

J H.

Numerade Educator

In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

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

J H.

Numerade Educator

In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

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

J H.

Numerade Educator

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

J H.

Numerade Educator

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

J H.

Numerade Educator

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

J H.

Numerade Educator

In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $.

(a) $ f(x) = \frac{1}{2} x^2 $ (b) $ g(x) = -\frac{1}{8} x^2 $

(c) $ h(x) = \frac{3}{2} x^2 $ (d) $ k(x) = -3x^2 $

J H.

Numerade Educator

In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $.

(a) $ f(x) = x^2 + 1 $ (b) $ g(x) = x^2 - 1 $

(c) $ h(x) = x^2 + 3 $ (d) $ k(x) = x^2 - 3 $

J H.

Numerade Educator

In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $.

(a) $ f(x) = (x - 1)^2 $ (b) $ g(x) = (3x)^2 + 1 $

(c) $ h(x) = \left(\frac{1}{3} x^2 \right) - 3 $ (d) $ k(x) = (x + 3)^2 $

J H.

Numerade Educator

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

(b) $ g(x) = \left[\frac{1}{2} (x -1) \right]^2 - 3 $

(c) $ h(x) = -\frac{1}{2} (x +1)^2 - 1 $

(d) $ k(x) = [2(x + 1)]^2 +4 $

Suzanne W.

Numerade Educator

In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).

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

J H.

Numerade Educator

In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).

$ g(x) = x^2 - 8 $

J H.

Numerade Educator

In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).

$ f(x) = x^2 + 7 $

J H.

Numerade Educator

$ h(x) = 12 - x^2 $

J H.

Numerade Educator

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

J H.

Numerade Educator

$ f(x) = 16 - \frac{1}{4} x^2 $

J H.

Numerade Educator

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

J H.

Numerade Educator

$ f(x) = (x - 6)^2 + 8 $

J H.

Numerade Educator

$ h(x) = x^2 - 8x + 16 $

J H.

Numerade Educator

$ g(x) = x^2 + 2x + 1 $

J H.

Numerade Educator

$ f(x) = x^2 - x + \frac{5}{4} $

J H.

Numerade Educator

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

J H.

Numerade Educator

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

J H.

Numerade Educator

$ f(x) = - x^2 - 4x + 1 $

J H.

Numerade Educator

$ h(x) = 4x^2 - 4x + 21 $

J H.

Numerade Educator

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

J H.

Numerade Educator

$ f(x) = \frac{1}{4} x^2 - 2x - 12 $

J H.

Numerade Educator

$ f(x) = -\frac{1}{3} x^2 + 3x - 6 $

Suzanne W.

Numerade Educator

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

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

J H.

Numerade Educator

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

$ f(x) = - (x^2 + x - 30) $

J H.

Numerade Educator

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

$ g(x) = x^2 + 8x + 11 $

J H.

Numerade Educator

$ f(x) = x^2 + 10x + 14 $

James K.

Numerade Educator

$ f(x) = 2x^2 - 16x + 31 $

J H.

Numerade Educator

$ f(x) = - 4x^2 + 24x - 41 $

Evelyn C.

Numerade Educator

$ g(x) = \frac{1}{2} (x^2 + 4x - 2) $

J H.

Numerade Educator

$ f(x) = \frac{3}{5} (x^2 + 6x - 5) $

Erik S.

Numerade Educator

In Exercises 43-46, write an equation for the parabola in standard form.

J H.

Numerade Educator

In Exercises 43-46, write an equation for the parabola in standard form.

J H.

Numerade Educator

In Exercises 43-46, write an equation for the parabola in standard form.

J H.

Numerade Educator

In Exercises 43-46, write an equation for the parabola in standard form.

J H.

Numerade Educator

In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point.

Vertex: $ ( -2, 5 ) $; point: $ ( 0, 9 ) $

J H.

Numerade Educator

In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point.

Vertex: $ ( 4, -1 ) $; point: $ ( 2, 3 ) $

Suzanne W.

Numerade Educator

In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point.

Vertex: $ ( 1, -2 ) $; point: $ ( -1, 14 ) $

J H.

Numerade Educator

Vertex: $ ( 2, 3 ) $; point: $ ( 0, 2 ) $

J H.

Numerade Educator

Vertex: $ ( 5, 12 ) $; point: $ ( 7, 15 ) $

J H.

Numerade Educator

Vertex: $ ( -2, -2 ) $; point: $ ( -1, 0 ) $

J H.

Numerade Educator

Vertex: $ \left(-\frac{1}{4}, \frac{3}{2} \right) $; point: $ ( -2, 0 ) $

J H.

Numerade Educator

Vertex: $ \left(\frac{5}{2}, -\frac{3}{4} \right) $; point: $ ( -2, 4 ) $

J H.

Numerade Educator

Vertex: $ \left(-\frac{5}{2}, 0 \right) $; point: $ \left(-\frac{7}{2}, -\frac{16}{3} \right) $

J H.

Numerade Educator

Vertex: $ ( 6, 6 ) $; point: $ \left(\frac{61}{10}, \frac{3}{2} \right) $

J H.

Numerade Educator

In Exercises 57 and 58, determine the x-intercept(s) of the graph visually. Then find the x-intercept(s) algebraically to confirm your results.

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

J H.

Numerade Educator

In Exercises 57 and 58, determine the x-intercept(s) of the graph visually. Then find the x-intercept(s) algebraically to confirm your results.

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

J H.

Numerade Educator

In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when $ f(x) = 0 $.

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

J H.

Numerade Educator

In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when $ f(x) = 0 $.

$ f(x) = -2x^2 + 10x $

J H.

Numerade Educator

In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when $ f(x) = 0 $.

$ f(x) = x^2 - 9x + 18 $

J H.

Numerade Educator

$ f(x) = x^2 - 8x - 20 $

J H.

Numerade Educator

$ f(x) = 2x^2 - 7x - 30 $

J H.

Numerade Educator

$ f(x) = \frac{7}{10} (x^2 + 12x -45) $

Jeremy S.

Numerade Educator

In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)

$ ( -1, 0 ) $, $ ( 3, 0 ) $

J H.

Numerade Educator

In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)

$ ( -5, 0 ) $, $ ( 5, 0 ) $

J H.

Numerade Educator

In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)

$ ( 0, 0 ) $, $ ( 10, 0 ) $

J H.

Numerade Educator

$ ( 4, 0 ) $, $ ( 8, 0 ) $

J H.

Numerade Educator

$ ( -3, 0 ) $, $ \left(-\frac{1}{2}, 0 \right) $

J H.

Numerade Educator

$ \left(-\frac{5}{2}, 0 \right) $, $ ( 2, 0 ) $

J H.

Numerade Educator

In Exercises 71-74, find two positive real numbers whose product is a maximum.

The sum is $ 110 $.

J H.

Numerade Educator

In Exercises 71-74, find two positive real numbers whose product is a maximum.

The sum is $ S $.

J H.

Numerade Educator

In Exercises 71-74, find two positive real numbers whose product is a maximum.

The sum of the first and twice the second is $ 24 $.

J H.

Numerade Educator

In Exercises 71-74, find two positive real numbers whose product is a maximum.

The sum of the first and three times the second is $ 42 $.

J H.

Numerade Educator

The path of a diver is given by

$ y = - \frac{4}{9} x^2 + \frac{24}{9} + 12 $

where $ y $ is the height (in feet) and $ x $ is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?

J H.

Numerade Educator

The height $ y $ (in feet) of a punted football is given by

$ y = - \frac{16}{2025} x^2 + \frac{9}{5} x + 1.5 $

where $ x $ is the horizontal distance (in feet) from the point at which the ball is punted.

(a) How high is the ball when it is punted?

(b) What is the maximum height of the punt?

(c) How long is the punt?

J H.

Numerade Educator

A manufacturer of lighting fixtures has daily production costs of $ C = 800 - 10x + 0.25x^2 $, where $ C $ is the total cost (in dollars) and $ x $ is the number of units produced. How many fixtures should be produced each day to yield a minimum cost?

J H.

Numerade Educator

The profit $ P $ (in hundreds of dollars) that a company makes depends on the amount $ x $ (in hundreds of dollars) the company spends on advertising according to the model $ P = 230 + 20x - 0.5x^2 $. What expenditure for advertising will yield a maximum profit?

J H.

Numerade Educator

The total revenue $ R $ earned (in thousands of dollars) from manufacturing handheld video games is given by

$ R(p) = -25p^2 + 1200p $

where $ p $ is the price per unit (in dollars).

(a) Find the revenues when the price per unit is $ \$ 20 $, $ \$25 $, and $ \$30 $.

(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain

your results.

Jeremy S.

Numerade Educator

The total revenue $ R $ earned per day (in dollars) from a pet-sitting service is given by $ R(p) = - 12 p^2 + 150p $, where $ p $ is the price charged per pet (in dollars).

(a) Find the revenues when the price per pet is $ \$4 $, $ \$6 $, and $ \$8 $ .

(b) Find the price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

J H.

Numerade Educator

A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).

(a) Write the area $ A $ of the corrals as a function of $ x $.

(b) Create a table showing possible values of $ x $ and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area

(c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area.

(d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area.

(e) Compare your results from parts (b), (c), and (d).

Suzanne W.

Numerade Educator

An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter single-lane running track.

(a) Draw a diagram that illustrates the problem. Let $ x $ and $ y $ represent the length and width of the rectangular region, respectively.

(b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of $ y $ around the inside edge of each semicircular part of the track.

(c) Use the result of part (b) to write an equation, in terms of $ x $ and $ y $, for the distance traveled in one lap around the track. Solve for $ y $.

(d) Use the result of part (c) to write the area of $ A $ the rectangular region as a function of $ x $ What dimensions will produce a rectangle of maximum area?

Tanya H.

Numerade Educator

A small theater has a seating capacity of 2000. When the ticket price is $ \$20 $, attendance is 1500. For each $ \$1 $ decrease in price, attendance increases by 100.

(a) Write the revenue $ R $ of the theater as a function of ticket price $ x $.

(b) What ticket price will yield a maximum revenue? What is the maximum revenue?

Harmendra S.

Numerade Educator

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 16 feet.

(a) Write the area $ A $ of the window as a function of $ x $.

(b) What dimensions will produce a window of maximum area?

J H.

Numerade Educator

From 1950 through 2005, the per capita consumption $ C $ of cigarettes by Americans (age 18 and older) can be modeled by $ C = 3565.0 + 60.30t - 1.783t^2, 0 \le t \le 55 $, where $ t $ is the year, with $ t = 0 $ corresponding to 1950.

(a) Use a graphing utility to graph the model.

(b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning

had any effect? Explain.

(c) In 2005, the U.S. population (age 18 and over) was 296,329,000. Of those, about 59,858,458 were

smokers. What was the average annual cigarette consumption per smoker in 2005? What was the

average daily cigarette consumption per smoker?

Sheryl E.

Numerade Educator

The sales $ y $ (in billions of dollars) for Harley-Davidson from 2000 through 2007 are shown in the table.

(a) Use a graphing utility to create a scatter plot of the data. Let $ x $ represent the year, with $ x = 0 $ corresponding to 2000.

(b) Use the regression feature of the graphing utility to find a quadratic model for the data.

(c) Use the graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data?

(d) Use the trace feature of the graphing utility to approximate the year in which the sales for Harley-Davidson were the greatest.

(e) Verify your answer to part (d) algebraically.

(f) Use the model to predict the sales for Harley-Davidson in 2010.

Check back soon!

In Exercises 87- 90, determine whether the statement is true or false. Justify your answer.

The function given by $ f(x) = -12x^2 - 1 $ has no x- intercepts.

J H.

Numerade Educator

In Exercises 87- 90, determine whether the statement is true or false. Justify your answer.

The graphs of $ f(x) = -4x^2 - 10x + 7 $ and $ g(x) = 12x^2 + 30x + 1 $ have the same axis of symmetry.

J H.

Numerade Educator

In Exercises 87- 90, determine whether the statement is true or false. Justify your answer.

The graph of a quadratic function with a negative leading coefficient will have a maximum value at its vertex.

J H.

Numerade Educator

In Exercises 87- 90, determine whether the statement is true or false. Justify your answer.

The graph of a quadratic function with a positive leading coefficient will have a minimum value at its vertex.

J H.

Numerade Educator

In Exercises 91- 94, find the values of such that the function has the given maximum or minimum value.

$ f(x) = - x^2 + bx - 75 $; Maximum :25

J H.

Numerade Educator

In Exercises 91- 94, find the values of such that the function has the given maximum or minimum value.

$ f(x) = -x^2 + bx - 16 $; Maximum: 48

J H.

Numerade Educator

In Exercises 91- 94, find the values of such that the function has the given maximum or minimum value.

$ f(x) = x^2 + bx + 26 $; Minimum value: 10

J H.

Numerade Educator

$ f(x) = x^2 + bx - 25 $; Minimum value: -50

J H.

Numerade Educator

Write the quadratic function

$ f(x) = ax^2 + bx + c $

in standard form to verify that the vertex occurs at

$ \left (-\frac{b}{2a} , f \left (-\frac{b}{2a} \right) \right) $.

J H.

Numerade Educator

The profit $ P $ (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the form

P = at^2 + bt + c

where $ t $ represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning.

(a)$ a $ is positive and $ -b / (2a) \le t $.

(b)$ a $ is positive and $ t \le -b/(2a) $.

(c)$ a $ is negative and $ -b /(2a) \le t $.

(d)$ a $ is negative and $ t \le -b(2a) $.

J H.

Numerade Educator

(a) Graph $ y = ax^2 $ for $ a = -2, -1, -0.5, 0.5, 1 $ and $ 2 $. How does changing the value of affect the graph?

(b) Graph $ y = (x - h)^2 $ for $ h= -4, -2, 2, $ and $ 4 $. How does changing the value of $ h $ affect the graph?

(c) Graph $ y = x^2 + k $ for $ k = -4, -2, 2, $ and $ 4 $. How does changing the value of $ k $ affect the graph?

J H.

Numerade Educator

Describe the sequence of transformation from to given that $ f(x) = x^2 $ and $ g(x) = a(x - h)^2 + k $. (Assume a, h, and k are positive.)

J H.

Numerade Educator

Is it possible for a quadratic equation to have only one $ x $ -intercept? Explain.

J H.

Numerade Educator

Assume that the function given by

$ f(x) = ax^2 + bx + c $, $ a \neq 0 $

has two real zeros. Show that the $ x $ -coordinate of the

vertex of the graph is the average of the zeros of $ f $.

(Hint: Use the Quadratic Formula.)

J H.

Numerade Educator