GRAPHICAL ANALYSIS In Exercises 5154, (a) use a graphing utility to graph the function

Fill in the blanks. The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps.

Fill in the blanks. The ________ ________ ________ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function.

Fill in the blanks. Polynomial functions of the form ________ are often referred to as power functions.

Fill in the blanks. A polynomial function of degree has at most ________ real zeros and at most ________ turning points.

Fill in the blanks. If is a zero of a polynomial function then the following three statements are true. (a) is a ________ of the polynomial equation (b) ________ is a factor of the polynomial (c) is an ________ of the graph of

Fill in the blanks. If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at and if it is of odd multiplicity, then the graph of ________ the -axis at

Fill in the blanks. A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right.

Fill in the blanks. The ________ ________ Theorem states that if is a polynomial function such that then, in the interval takes on every value between and _a, b_, f f _a_ f _b_.

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 916, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and (h).]

In Exercises 1720, sketch the graph of and each transformation.

In Exercises 1720, sketch the graph of and each transformation.

In Exercises 1720, sketch the graph of and each transformation.

In Exercises 1720, sketch the graph of and each transformation.

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

In Exercises 2130, describe the right-hand and left-hand behavior of the graph of the polynomial function

GRAPHICAL ANALYSIS In Exercises 3134, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

GRAPHICAL ANALYSIS In Exercises 3134, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

GRAPHICAL ANALYSIS In Exercises 3134, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

GRAPHICAL ANALYSIS In Exercises 3134, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 3550, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

GRAPHICAL ANALYSIS In Exercises 5154, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

GRAPHICAL ANALYSIS In Exercises 5154, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

GRAPHICAL ANALYSIS In Exercises 5154, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

GRAPHICAL ANALYSIS In Exercises 5154, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 55 64, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6574, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 7588, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 8992, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8992, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8992, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8992, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 9396, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results.

In Exercises 9396, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results.

In Exercises 9396, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results.

In Exercises 9396, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results.

NUMERICAL AND GRAPHICAL ANALYSIS An open box is to be made from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length from the corners and turning up the sides (see figure). (a) Write a function that represents the volume of the box. (b) Determine the domain of the function. (c) Use a graphing utility to create a table that shows box heights and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. (d) Use a graphing utility to graph and use the graph to estimate the value of for which is maximum. Compare your result with that of part (c).

MAXIMUM VOLUME An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure. (a) Write a function that represents the volume of the box. (b) Determine the domain of the function (c) Sketch a graph of the function and estimate the value of for which is maximum.

CONSTRUCTION A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure). (a) Let represent the height of the sidewall of the gutter. Write a function that represents the cross-sectional area of the gutter. (b) The length of the aluminum sheeting is 16 feet. Write a function that represents the volume of one run of gutter in terms of (c) Determine the domain of the function in part (b). (d) Use a graphing utility to create a table that shows the sidewall heights and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. (e) Use a graphing utility to graph Use the graph to estimate the value of for which is a maximum. Compare your result with that of part (d). (f) Would the value of change if the aluminum sheeting were of different lengths? Explain.

CONSTRUCTION An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components (see figure). (a) Write a function that represents the total volume of the tank in terms of (b) Find the domain of the function. (c) Use a graphing utility to graph the function. (d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of the tank.

REVENUE The total revenues (in millions of dollars) for Krispy Kreme from 2000 through 2007 are shown in the table. A model that represents these data is given by where represents the year, with corresponding to 2000. (Source: Krispy Kreme) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use a graphing utility to approximate any relative extrema of the model over its domain. (d) Use a graphing utility to approximate the intervals over which the revenue for Krispy Kreme was increasing and decreasing over its domain. (e) Use the results of parts (c) and (d) to write a short paragraph about Krispy Kremes revenue during this time period.

REVENUE The total revenues (in millions of dollars) for Papa Johns International from 2000 through 2007 are shown in the table. A model that represents these data is given by where represents the year, with corresponding to 2000. (Source: Papa Johns International) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use a graphing utility to approximate any relative extrema of the model over its domain. (d) Use a graphing utility to approximate the intervals over which the revenue for Papa Johns International was increasing and decreasing over its domain. (e) Use the results of parts (c) and (d) to write a short paragraph about the revenue for Papa Johns International during this time period.

TREE GROWTH The growth of a red oak tree is approximated by the function where is the height of the tree (in feet) and is its age (in years). (a) Use a graphing utility to graph the function. (Hint: Use a viewing window in which and (b) Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year. (c) Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by Find the vertex of this parabola. (d) Compare your results from parts (b) and (c).

REVENUE The total revenue (in millions of dollars) for a company is related to its advertising expense by the function where is the amount spent on advertising (in tens of thousands of dollars). Use the graph of this function, shown in the figure on the next page, to estimate the point on the graph at which the function is increasing most rapidly. This point is called the point of diminishing returns because any expense above this amount will yield less return per dollar invested in advertising.

A fifth-degree polynomial can have five turning points in its graph.

It is possible for a sixth-degree polynomial to have only one solution.

The graph of the function given by rises to the left and falls to the right.

CAPSTONE For each graph, describe a polynomial function that could represent the graph. (Indicate the degree of the function and the sign of its leading coefficient.)

GRAPHICAL REASONING Sketch a graph of the function given by Explain how the graph of each function differs (if it does) from the graph of each function Determine whether is odd, even, or neither. (a) (b) (c) (d) (e) (f ) (g) (h)

THINK ABOUT IT For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and lefthand behavior of the graph of the function. (a) (b) (c) (d) (e) (f ) (g)

THINK ABOUT IT Sketch the graph of each polynomial function. Then count the number of zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) (b) (c)

Explore the transformations of the form (a) Use a graphing utility to graph the functions and Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of always be increasing or decreasing? If so, is this behavior determined by or Explain. (c) Use a graphing utility to graph the function given by Use the graph and the result of part (b) to determine whether can be written in the form Explain.