algebra 2 honors: quadratic functions 2 honors: quadratic functions semester 1, ... summation...

Algebra 2 Honors: Quadratic Functions Semester 1, Unit 3: Activity 14

Resources:

SpringBoard- Algebra 2

Online Resources:

Algebra 2 Springboard

Text

Unit 3 Vocabulary:

Alternative Polynomial function Degree Standard form of a polynomial Relative maximum Relative minimum End behavior Even function Odd function Synthetic division Combination Factorial Summation notation Fundamental Theorem of Algebra Extrema Relative extrema Global extrema

Unit Overview

In this unit, students begin by writing and graphing a third-degree equation that represents a real-world situation. They perform operations on polynomials; factor polynomials; identify the extrema, zeros, and roots of polynomials; and study the end behavior of graphs of polynomial functions.

Student Focus

Main Ideas for success in lessons 14-1, 14-2 & 14-3:

Write a third-degree equation representing a real-world situation

Graph a portion of a third-degree equation

Identify the relative minimum and maximum of third-degree equations

Examine end behavior of polynomial functions

Determine even and odd functions using algebraic and geometric techniques

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Example:

Lesson 14-1: Angie is determining the volume of a box in the shape of a rectangular prism. She finds the length is 2 cm more than the width and 3 cm less than the depth. Which equation can Angie use to get the volume of the box, in cubic centimeters?

Lesson 14-2:

Polynomial Functions

Degree Even or Odd?

Sign of leading

coefficient End Behavior

2 Even + same direction at both ends (+)

2 Even - same direction at both ends (-)

3 Odd + opposite direction at either end

( )

3 Odd - opposite direction at either end

( )

4 Even + same direction at both ends (+)

4 Even - same direction at both ends (-)

5 Odd + opposite direction at either end

5 Odd - opposite direction at either end

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Lesson 14-3 Meghan examines several polynomial functions and thinks that she has found a pattern. She believes that every even-degree polynomial is even and that every odd-degree polynomial is odd. Which function disproves her assertion? Why?

The degree of the function is odd, 3, however as seen in the graph, the

function is not symmetrical over the origin.

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Resources:


Online Resources:


Text

Unit 3 Vocabulary:


Unit Overview


Student Focus

Main Ideas for success in lessons 15-1, 15-2 & 15-3:

Perform operations with polynomials (including addition, subtraction, multiplication, long division, and synthetic division)

Example: Lesson 15-1: Maria has a business desigining and making custom T-shirts. Her cost for materials, in

thousands of dollars, is given by where t is the number of the month (1-12) on the last day of the month. Her income from sales is given by . What is the process for finding the function that gives her profit as a function of time and what does it equal?

substitution commutative property subtraction Lesson 15-2:

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Lesson 15-3 Example 1:

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Example 2:

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Resources:


Online Resources:


Text

Unit 3 Vocabulary:


Unit Overview


Student Focus

Main Ideas for success in lessons 16-1, & 16-2:

The binomial theorem and how to use it for binomial expansion

Apply the binomial theorem to identify terms and coefficients of a binomial expansion

Example: Lesson 16-1:

Example 1:

Pascal’s Triangle (first 5 rows):

Example 2:

Which statement best describes the relationship between the binomial and Pascal’s triangle?

A. The coefficients of the expanded binomial are equal to the numbers in the second row of Pascal’s triangle

B. The coefficients of the expanded binomial are equal to the numbers in the third row of Pascal’s triangle

C. The roots of the binomial are equal to the numbers in the second row of Pascal’s triangle

D. The roots of the binomial are equal to the numbers in the third row of Pascal’s triangle

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Lesson 16-2: Binomial Theorem:

Example:

Use the binomial theorem to determine the sixth term of ordered by the

exponents of a from largest to smallest.

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Resources:


Online Resources:


Text

Unit 3 Vocabulary:


Unit Overview


Student Focus

Main Ideas for success in lessons 17-1, & 17-2:

Factor higher-order polynomials

Factor using various techniques such as factoring trinomials, the sum or difference of squares or cubes, and by grouping

Know and apply the Fundamental Theorem of Algebra

Write polynomial functions

Example: Lesson 17-1: Example (factoring a trinomial):

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Example (factoring by grouping):

Difference of Squares:

Lesson 17-2: Example (Fundamental Theorem of Algebra):

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Example (writing polynomial functions):

Example (Complex Conjugate Root Theorem):

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Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 3

Vocabulary:

Alternative Polynomial function Degree Standard form of a polynomial Relative maximum Relative minimum End behavior Even function Odd function Synthetic division Combination Factorial Summation notation Fundam. Theorem of Algebra Extrema Relative extrema Global extrema

Unit Overview


Student Focus

Main Ideas for success in lessons 18-1, 18-2, & 18-3:

Graph polynomial functions

Describe roots of polynomial functions

Compare properties of functions represented in different ways

Use graphing to solve polynomial inequalities

Example: Lesson 18-1:

k(x) = x4 − 10x2 + 9 = (x + 3)(x − 3)(x + 1)(x − 1) The unfactored polynomial reveals that the function is even (degree = 4), so the graph is symmetrical around the y-axis. The leading coefficient is positive ( ), so the value of the function increases as x approaches negative and positive infinity ( ). It also reveals that the y-intercept is 9. The factored form shows that the x-intercepts are −3, −1, 1, and 3.

Vocabulary representations:

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Lesson 18-2:

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Lesson 18-3:

Example (comparing functions):

The function f(x) is a quadratic function. The -term has a positive coefficient, and

the vertex of f(x) is at (–2, –2).

The function g(x) is given by the equation Which

function has a greater range, and why?

A. f(x), because the highest-degree term has a positive coefficient in f(x) and a

negative coefficient in g(x) B. f(x), because as x→∞,y→∞, whereas with g(x), as x→∞,y→−∞

C. g(x), because the range of g(x) is (−∞,∞), whereas with f(x), the range

is [−2,∞)

D. g (x), because it is a higher-degree polynomial than f(x)

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1© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

LeSSon 14-1 1. Model with mathematics. The volume of a

rectangular box is given by the expression V 5 (160 2 4w)w2.

a. What is a reasonable domain for the function in this situation?

b. Sketch a graph of the function over the domain that you found. Include the scales on each axis.

c. Approximate the coordinates of the maximum point of the function.

d. What is the width of the box at the maximum volume?

2. A cylindrical package is being designed for a new product. The height of the package plus twice its radius must be less than 30 in.

a. Write an expression for h, the height of the tube, in terms of r, the radius of the tube.

b. Write an expression for V, the volume of the tube, in terms of r, the radius of the tube.

c. Find the radius that yields the maximum value.

d. Find the maximum volume of the tube.

3. Why is the sketch of the graphs of V(w) in Items 1 and 2 limited to the first quadrant?

4. The volume of a package V is a function of w, the width of the square ends of the package such that V(w) 5 (180 2 4w)w2. Which of the following is the domain of the function?

A. 0 # w # 45 B. 0 , w , 45

C. 0 # w # 453 D. 0 , w , 453

5. Reason abstractly. Why is the volume function of a prism or cylinder a third-degree equation?

LeSSon 14-2 6. Decide if the function f (x) 5 28x2 1 7x3 1 2x 2 5

is a polynomial. If it is, write the function in standard form and then state the degree and leading coefficient.

7. Which of the following is NOT a polynomial?

A. f (x) 5 3x3 2 5x2 1 7x 2 11

B. f (x) 5 12

x2 1 5x3 2 8

C. f (x) 5 0.86x 2 6x4 1 3x2 2 5x3 1 9

D. f (x) 5 23x5 1 2x3 2 8x12 1 6

Algebra 2 Unit 3 Practice

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© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

8. Examine the graph.

x

y

25

25

5

5

a. Describe the leading coefficient.

b. Describe the end behavior of the function.

c. Name any x- or y-intercept(s) of the function.

d. Name any relative maximum values and minimum values of the function.

9. Use appropriate tools.

a. Sketch the graph of f (x) 5 0.5x3 2 3x2 1 1.

b. Name any x- or y-intercept(s) of the function.

c. Name any relative maximum values and relative minimum values of the function.

10. Use a graphing calculator to determine the minimum number of times a cubic (third-degree) function must cross the x-axis and the maximum number of times it can cross the x-axis.

LeSSon 14-3 11. Determine algebraically if each function is even or

odd by substituting f (2x) for f (x). Show your work.

a. f (x) 5 2x2 1 5

b. f (x) 5 23x3 1 2x

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12. Determine whether the function is even, odd, or neither. Justify your answer.

x

y

25

25

5

5

13. Attend to precision. Give an example of a polynomial function that has an even degree but is not an even function. Explain.

14. For a given polynomial function, as x → `, the graph increases without bound, and as x → -`, the graph decreases without bound. Is this an odd or even function? Explain.

15. If f (x) is an even function and passes through the point (22, 7), which other point must lie on the graph of the function?

A. (22, 27) B. (2, 27)C. (2, 7) D. (7, 22)

LeSSon 15-1 16. Bruce owns Bruce’s Bakery and his daughter

Hannah owns Hannah’s Cakes. The function B(t) 5 t3 2 17t2 1 68t 1 51 represents the number of cakes Bruce’s Bakery sold each month last year, and the function H(t) 5 t3 2 18t2 1 75t 1 65 represents the number of cakes Hannah’s Cakes sold each month last year. The variable t represents the number of the month (1212) on the last day of the month.

a. In January, how many cakes did the two bakeries sell altogether?

b. Which bakery sold more cakes in January? How many more?

17. The function R(t) represents Bruce’s revenue from the sale of both cakes and pastries. The revenue function for cakes is C(t) 5 27t3 2 420t2 1 1400t 1 2000 and the revenue function for pastries is P(t) 5 45t3 2 820t2 1 4200t 2 1500, where t represents the number of the month (1212) in that year.

a. Write the revenue function R(t) that represents Bruce’s total revenue from both cakes and pastries.

b. Use appropriate tools. Use a graphing calculator to graph all three revenue functions on the same coordinate plane. Determine when the relative maximum(s) and minimum(s) occur for each function.

c. Compare the value of R(t) to the value of C(t) and P(t) for every t.

d. How can you find the value of C(t) if you know the value of R(t) and the value of P(t)?

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18. The polynomial expressions 3x 1 5, 4x2 2 7x, 5x 1 1, and 2x2 1 13x represent the lengths of the sides of a quadrilateral for all whole-number values of x . 1. Which is the expression for the perimeter of the quadrilateral?

A. 4x2 1 x 1 6 B. 6x2 1 14x 1 6

C. 10x2 1 8x 1 6 D. 14x2 1 6x 1 6

19. Mari makes and sells handbags. The cost and sales of the handbags are seasonal. Mari’s revenue R(t) and cost function C(t) are shown below. t represents the number of the month (1212).

R(t) 5 51t3 2 892t2 1 5400t 1 200

C(t) 5 65t3 2 770t2 1 990t 1 9000

a. Use a graphing calculator to graph the revenue and cost functions on the same coordinate plane. What is the domain?

b. Does Mari ever experience a loss during the year? If so, when? What happens in December?

c. When is the break-even point? What is the revenue and the cost at that point?

d. Write the profit function for Mari’s handbag business. Use a graphing calculator to graph the profit function on the same coordinate plane as the revenue and cost functions.

e. What is the profit at the break-even point? Where is this point on the graph of the profit function?

f. When does Mari experience the greatest profit? What is this point on the graph called?

g. When is the profit negative? Why does this occur?

20. If you ran a business and found that during certain months of the year the business was running at a loss, what might you do?

LeSSon 15-2 21. Find each sum or difference. Write your answer as

a polynomial in standard form.

a. (5x4 1 7x3 1 2x2 1 25x 2 9) 1 (6x4 2 9x3

1 8x2 2 17x 1 5)

b. (9x5 2 5x3 1 12x 1 8) 2 (7x5 1 3x4 2 7x2 1 5x 2 12)

c. (21x4 1 3x2 2 7) 2 (25x2 1 9x 1 8)

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d. (8x2 1 7x 1 18) 1 (3x3 2 11) 2 (5x3 2 8x2 2 3x)

e. (27x3 1 5x2 2 9) 2 (6x3 1 7x2 2 2x) 2 (3x2 1 15)

22. Find each product. Write your answer as a polynomial in standard form.

a. (x 2 3)(x2 1 9x 2 2)

b. (x3 2 5x 1 13)(3x 2 2)

c. (5x2 2 7)(5x2 2 4x 1 9)

d. (7x2 2 9x 1 11)(2x2 2 5x 1 4)

23. Attend to precision. Which polynomial is (x 2 2)3?

A. 3x2 2 6

B. x6 2 8

C. 3x6 2 6x2 2 8

D. x3 2 6x2 1 12x 2 8

24. What type of expression is each sum, difference, or product in Items 21223?

25. An open box is made by cutting four squares of equal size from the corners of a 12-inch-by-16-inch rectangular piece of cardboard and then folding up the sides.

a. What expression can be used to find the volume?

b. Write the volume of the box as a polynomial in standard form.

LeSSon 15-3 26. Use long division to find each quotient.

a. (x3 1 5x2 2 7x 2 35) 4 (x2 2 7)

b. (x3 2 5x2 1 x 2 5) 4 (x 2 5)

c. (x3 1 3x2 2 7x 1 9) 4 (x 1 3)

d. (5x3 2 10x2 1 15x) 4 (x2 2 2x 1 3)

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27. Use synthetic division to find each quotient.

a. (x4 1 5x3 1 x2 2 14x 1 3) 4 (x 1 3)

b. (x3 2 2x2 1 9x 2 18) 4 (x 2 2)

c. (x3 1 2x2 2 13x 1 10) 4 (x 1 5)

d. (2x5 2 8x4 1 5x3 2 20x2 2 7x 1 28) 4 (x 2 4)

28. Which form should the divisor have in order for synthetic division to be useful?

A. x 1 k B. x 2 k

C. x2 1 k D. x2 2 k

29. Make use of structure. The product of two polynomials is x3 1 11x2 1 13x 2 10. One factor is x 1 2.

a. Would you use long division or synthetic division to find the other factor? Explain.

b. What is the other factor?

30. Write the steps for finding the quotient of (6x3 1 x 2 1) 4 (x 1 20) using synthetic division.

LeSSon 16-1 31. Evaluate each combination.

a. 12C8

b. 6C2

c. 2015

d. 1812

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32. Use appropriate tools. Use a graphing calculator to determine how many different combinations of five scientists can be selected from a group of 15.

33. Which row of numbers represents the fifth row of Pascal’s Triangle?

A. 1 3 3 1

B. 1 4 6 4 1

C. 1 5 10 10 5 1

D. 1 6 15 20 15 6 1

34. Expand (a 1 b)4.

35. How does the number of terms in the expansion of (a 1 b)5 relate to the exponents? How many terms are there?

LeSSon 16-2 36. express regularity in repeated reasoning. Find the

coefficient of the specified term in each expansion.

a. the third term in (x 1 5)6

b. the second term in (x 1 3)5

c. the fifth term in (x 1 6)10

d. the sixth term in (x 2 4)8

37. Find the specified term of each expansion.

a. the fourth term in (x 2 3)7

b. the fifth term in (x 1 7)16

c. the eighth term in (2x 2 5)10

d. the seventh term in (3x 1 8)9

38. Use the binomial theorem to write the binomial expansion of (x 2 3)5.

39. Which of the following is the sixth term in the expansion of (x 2 9)7?

A. 21,240,029x B. 21,240,029x2

C. 21,240,029x5 D. 21,240,029x6

40. Write and evaluate the expression n

r 1

2

an 2 (r 2 1)br 2 1 for n 5 5, r 5 3, a 5 2x,

and b 5 4.

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LeSSon 17-1 41. Factor each polynomial.

a. x2 2 7x 1 12

b. 3x2 1 x 2 10

c. 3x4 1 2x2 2 5

d. x2 1 5x 2 36

42. Factor by grouping.

a. 2x3 2 6x2 1 5x 2 15

b. 3x4 2 x3 1 6x 2 2

c. x3 1 5x2 2 9x 2 45

d. x3 2 5x2 2 3x 1 15

43. Factor each sum or difference of cubes.

a. x3 1 125

b. x3 2 8

c. 8x3 1 216

d. 64x3 2 27

44. Use the formulas for factoring quadratic binomials and trinomials to factor each expression.

a. 25x4 2 169

b. x4 1 6x2 1 9

c. x6 2 10x3 1 25

d. 4x10 2 81

45. Make use of structure. Which of the following are the factors of 27x3 2 8?

A. (3x 2 2)(9x2 2 6x 2 4)

B. (3x 2 2)(9x2 1 6x 2 4)

C. (3x 2 2)(9x2 2 6x 1 4)

D. (3x 2 2)(9x2 1 6x 1 4)

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LeSSon 17-2 46. Find the zeros of the functions. Show that the

Fundamental Theorem of Algebra is true for each function by counting the number of complex zeros.

a. f (x) 5 x3 1 4x

b. g(x) 5 x4 2 81

c. h(x) 5 2x4 2 16x3 1 32x2

d. j(x) 5 4x3 2 4x2 2 x 1 1

47. Attend to precision. Complete the following statement:

As a consequence of the Fundamental Theorem of Algebra,

48. Write a polynomial function of nth degree that has the given real roots.

a. n 5 3; zeros: 21, 0, 2

b. n 5 4; zeros: 23, 2, 61

c. n 5 3; x 5 21, and x 5 3 is a double root

d. n 5 4; x5 22 is a double root, and x 5 5 is a double root

49. Which is the degree of the polynomial function

with the roots x 5 23, x 5 23

, x 5 i, and x 5 1 – i?

A. 4 B. 5

C. 6 D. 8

50. Write a polynomial function of nth degree that has the given real or complex roots.

a. n 5 3; x 5 2, x 5 3i

b. n 5 4; x 5 2i, x 5 1 1 3i

c. n 5 5; x 5 0 is a double root, x 5 3, x 5 2 2 i

d. n 5 4; x 5 12

i, x 5 612

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LeSSon 18-1 51. Match each equation to its graph.

a. f (x) 5 213

x 2 2

b. g(x) 5 x2 1 3

c. h(x) 5 2x3 2 3x2 1 1

d. j(x) 5 12

x4 2 3x2 1 4

e. k(x) 5 212

x5 2 x4 2 x3 2 x2 2 x

II.

x

y

25

25

5

10

5

IV.

x

y

2122

210

25

5

10

1 2

I.

x

y

25210

210

25

5

5 10

III.

x

y

25

210

25

5

10

5

V.

x

y

25

10

5

15

5

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52. For polynomials, relative extrema occur

A. where the graph crosses the y-axis.

B. at the ends of the polynomial function.

C. where the graph crosses the x-axis.

D. between the zeros of the polynomial function.

53. Use appropriate tools. Use a graphing calculator to graph the polynomial functions. Determine the coordinates of the intercepts and relative extrema.

a. f (x) 5 x4 1 x3 2 50x2 1 x

b. g(x) 5 2x3 2 19x2 2 48

54. Sketch the graph of a polynomial function that increases as x → ∞, decreases as x → 2∞, and has zeros at x 5 25, 0, and 2.

55. a. Use a graphing calculator to graph f (x) 5 2x4 1 3x3 1 5x2 2 10x 1 1.

b. Find all the intercepts of the function.

c. Find the relative extrema of the function.

LeSSon 18-2 56. Find all possible rational roots of each equation.

a. f (x) 5 3x3 1 5x2 2 4x 1 5

b. g(x) 5 2x4 1 7x3 2 3x2 1 5x 2 6

57. Given p(x) 5 2x5 1 6x4 2 3x3 2 5x2 1 3x 2 7:

a. How many sign changes are there?

b. How many possible positive real roots are there?

c. Find p(2x).

d. How many negative real roots are there?

58. Make use of structure. Find the number of positive and negative real roots of each equation. Explain.

a. h(x) 5 x3 2 x2 1 3x 1 5

b. j(x) 5 5x4 2 2x3 1 3x2 1 10x 2 5

59. Given k(x) 5 x3 2 2x2 2 5x 1 6:

a. Find the real zeros of k(x).

b. What is the y-intercept?

c. Find the relative maximum and minimum to the nearest integer.

d. Graph k(x) by hand.

x

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60. The graph of m(x) has an x-intercept at (23, 0). Which of the following is NOT true?

I. m(23) 5 0 II. x 1 3 is a factor of m(x). III. m(x) also has an x-intercept at (3, 0).

A. I only B. II only

C. III only D. II and III only

LeSSon 18-3 61. Which representation below is a quadratic function

that has zeros at x 5 2 and x 5 4? Explain.

A. h(x) 5 x2 2 6x 1 8

B. x 0 1 2 3 4y 24 22 0 2 4

62. The graph of q(x) is shown below. Use the graph to solve for q(x) # 0.

x

q(x)

25

25

5

10

5

63. Make sense of problems. The function m(x) is a polynomial that increases without bound as x → 6∞, has a double root at 22, and has no other real roots. The function p(x) is given by the equation p(x) 5 x2 2 25. Which function has the greater range? Explain your reasoning.

64. Solve each inequality.

a. (x 1 5)(x 2 1)(x 2 8) . 0

b. x3 2 5x2 2 2x 1 24 # 0

65. The graph of f (x) is shown. Which of the following inequalities is the solution of f (x) $ 0?

x

f(x)

2428

24

28

212

216

4

8

12

16

20

24

4 8

A. 2∞ # x $ ∞

B. 25 # x #21, and x $ 2

C. 25 # x #21 and x # 2

D. 2∞ # x #25, and 21 # x # 2

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algebra 2 honors: quadratic functions 2 honors: quadratic functions semester 1, ... summation...

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