Deck 3: Polynomial and Rational Functions

Question 1

Find a quadratic function whose graph is a parabola with - intercepts of and and lowest point .

Answer 1

In order to find , find the midpoint between the points and . . Since the function passes through the point , we can find by substituting these values in the equation . Thus the quadratic function is .

Question 2

Find a quadratic function whose graph is a parabola with - intercepts of and and highest point .

Answer 2

In order to find , find the midpoint between the points and . Since the function passes through the point , we can find by substituting these values in the equation Thus the quadratic function is .

Question 3

Sketch the graph of the parabola and state the coordinates of its vertex and its intercepts.

Answer 3

Vertex: , - intercepts: , - intercept:

Question 4

The cost associated with storing units of a certain product is given by the function where is measured in of dollars and in hundreds of units.
(a) For what values of
will the cost be decreasing? increasing?
(b) What is the minimum cost?

Question 5

Find the maximum or the minimum value of the function .

Question 6

If a ball is thrown directly upward with a velocity of 48 ft/s, its height (in feet) after t seconds is given
by . What is the maximum height attained by the ball?

Question 7

Sketch the graph of the parabola and state the coordinates of its vertex and its intercepts.

Question 8

Find the maximum or minimum value of the function .

Question 9

A quadratic function is given.
(a) Express the quadratic function in standard form.
(b) Find its vertex and its x- and y-intercept(s).

Question 10

Sketch the graph of the parabola and state the coordinates of its vertex and its intercepts.

Question 11

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, where the revenue is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Question 12

A quadratic function is given.
(a) Express the quadratic function in standard form.
(b) Find its vertex and its x- and y-intercept(s).

Question 13

If a ball is thrown directly upward with a velocity of 64 ft/s, its height (in feet) after t seconds is given
by . What is the maximum height attained by the ball?

Question 14

Find the minimum value of a parabola with - intercepts of and and - intercept .

Question 15

Find the maximum or minimum value of the function .

Question 16

Find two positive numbers whose sum is and the sum of whose squares is a minimum.

Question 17

Find the maximum or the minimum value of the function .

Question 18

Among all rectangles that have a perimeter of ft, find the dimensions of the one with the largest area.

Question 19

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, where the revenue is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Question 20

Find the maximum or minimum value of the function .

Question 21

Sketch the graph of the function by transforming the graph of an appropriate function of the form . Indicate all - and - intercepts on your graph.

Question 22

Sketch the graph of the function . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Question 23

Sketch the graph of the function . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Question 24

Let . Graph the family of functions with , , , , and in the viewing rectangle by . How does the value of affect the graph?

Question 25

Sketch the graph of the function . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Question 26

On the same coordinate axes, sketch the graphs (as accurately as possible) of the functions and . Based on your sketch, how many points do the graphs appear to intersect? Find the coordinates of all intersection points.

Question 27

Sketch the graph of the function by transforming the graph of an appropriate function of the form . Indicate all - and - intercepts on your graph.

Question 28

Give an example of a polynomial that has local extrema.

Question 29

Sketch the graph of the function and describe its end behavior.

Question 30

Use a graphing device to graph the polynomial and describe its end behavior.

Question 31

Sketch the graph of the function by transforming the graph of an appropriate function of the form . Indicate all - and - intercepts on your graph.

Question 32

A business consultant analyzing a small computer company finds that if the company produces and sells microprocessors it has a total profit function (in thousands of dollars) given by , with (in hundreds), being the number of processors produced and sold annually.
(a) Graph the profit function and determine how many processors must be produced and sold in order to break even.
(b) What is the domain of the profit function where the company actually loses money?
(c) Estimate the number of processors that must be produced in order to give the company the largest possible profit.
(d) Estimate the largest profit.

Question 33

Find the quotient and remainder of .

Question 34

Sketch the graph of the function . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Question 35

Let . Graph the family of functions with , , , , and in the viewing rectangle by . How does the value of affect the graph?

Question 36

An open box is to be constructed from a piece of sheet metal in by in by cutting squares of length from each corner and folding up the sides.
(a) Express the volume of the box as a function of
.
(b) What is the domain of ?
(c) Draw a graph of the function and use it to estimate the maximum volume for the box.

Question 37

Sketch the graph of the function by transforming the graph of an appropriate function of the form . Indicate all - and - intercepts and make sure your graph exhibits the proper end behavior.

Question 38

Graph the family of polynomials in the same viewing rectangle, using the values , , , , and . Explain how changing the value of affects the graph.

Question 39

Use a graphing device to graph the polynomial and describe its end behavior.

Question 40

Graph the family of polynomials in the same viewing rectangle, using the values , , , and . Explain how changing the value of affects the graph.

Question 41

Use the Factor Theorem to show that is a factor of the polynomial .

Question 42

Find the quotient and remainder of using synthetic division.

Question 43

Use synthetic division and the remainder theorem to evaluate for . 

Question 44

Let . Show that , and use this fact to find all the zeros of .  

Question 45

Find the quotient and remainder of using long division.

Question 46

Find the quotient and remainder of using synthetic division.

Question 47

Find the quotient and remainder of using synthetic division.

Question 48

Find the quotient and remainder using long division.

Question 49

Use the Factor Theorem to show that is a factor of the polynomial .

Question 50

Find the quotient and remainder using long division.

Question 51

Use synthetic division and the remainder theorem to evaluate for .

Question 52

Find the quotient and remainder of using long division.

Question 53

Use synthetic division and the remainder theorem to evaluate for .

Question 54

Find the quotient and remainder of .

Question 55

Find the quotient and remainder of using long division.

Question 56

Find the quotient and remainder of using long division.

Question 57

Use synthetic division and the remainder theorem to evaluate for .

Question 58

Find the quotient and remainder of .

Question 59

Find the quotient and remainder of using long division.

Question 60

Use the Factor Theorem to show that is a factor of the polynomial .

Question 61

Find all rational zeros of the polynomial .

Question 62

Find all rational zeros of the polynomial .

Question 63

Find all rational zeros of the polynomial .

Question 64

Find a polynomial of degree 3 that has zeros 1, 2, and -3 and in which the coefficient of is -2.

Question 65

Find all rational zeros of the polynomial .

Question 66

Find all rational zeros of the polynomial .

Question 67

Find the remainder if is divided by .

Question 68

Find all rational zeros of the polynomial .

Question 69

Find the polynomial of degree that has zeros , , and .  

Question 70

Find all rational zeros of the polynomial .

Question 71

Find a polynomial of degree 3 that has zeros 1, 2, and -5 and in which the coefficient of is 3.

Question 72

Find the polynomial of degree that has zeros , , , and .

Question 73

Find all rational zeros of the polynomial .

Question 74

Find all rational zeros of the polynomial .

Question 75

Find the remainder of , if is divided by . What is the significance of this remainder?

Question 76

Find a polynomial of degree 3 that has zeros 1, -2, and 3 and in which the coefficient of is -5.

Question 77

Find the polynomial of degree that has zeros , , and , with and being double roots.

Question 78

Let . Show that , and . Use this fact to find all the zeros of .

Question 79

Find all rational zeros of the polynomial .

Question 80

Find all rational zeros of the polynomial .