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

Online Resources: Algebra 2 Springboard Text

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 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

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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

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

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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 (

) opposite direction at either end

3

Odd

-

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

(

)

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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Algebra 2 Honors: Quadratic Functions Semester 1, Unit 3: Activity 15 Resources: SpringBoard- Algebra 2

Online Resources: Algebra 2 Springboard Text

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 15-1, 15-2 & 15-3:

Unit 3 Vocabulary:

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

Alternative Example: Polynomial function Lesson 15-1: Degree 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 Standard form of a month (1-12) on the last day of the month. Her income from sales is given by polynomial . What is the process for finding the function that gives her profit as a Relative maximum function of time and what does it equal? Relative minimum End behavior Even function Odd function Synthetic division Combination Factorial Summation notation Fundamental Theorem of substitution Algebra commutative property Extrema subtraction Relative extrema Global extrema Lesson 15-2:

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Lesson 15-3 Example 1:

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Example 2:

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Algebra 2 Honors: Quadratic Functions Semester 1, Unit 3: Activity 16 Resources: SpringBoard- Algebra 2

Online Resources: Algebra 2 Springboard Text

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 16-1, & 16-2:

Unit 3 Vocabulary:

 The binomial theorem and how to use it for binomial expansion  Apply the binomial theorem to identify terms and coefficients of a binomial expansion

Alternative Polynomial function Example: Degree Lesson 16-1: Standard form of a polynomial Relative maximum Relative minimum Example 1: End behavior Even function Odd function Synthetic division Combination Factorial Summation notation Pascal’s Triangle (first 5 rows): Fundamental Theorem of Algebra Extrema Relative extrema Global extrema

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 Page 7 of 27

Lesson 16-2: Binomial Theorem:

Example: Use the binomial theorem to determine the sixth term of exponents of a from largest to smallest.

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ordered by the

Algebra 2 Honors: Quadratic Functions Semester 1, Unit 3: Activity 17 Resources: SpringBoard- Algebra 2

Online Resources: Algebra 2 Springboard Text

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 17-1, & 17-2:

Unit 3 Vocabulary:

 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

Alternative Polynomial function Degree Example: Standard form of a polynomial Lesson 17-1: Relative maximum Relative minimum Example (factoring a trinomial): End behavior Even function Odd function Synthetic division Combination Factorial Summation notation Fundamental Theorem of Algebra Extrema Relative extrema Global extrema

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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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Algebra 2 Honors: Quadratic Functions Semester 1, Unit 3: Activity 18 Resources: SpringBoardAlgebra 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

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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 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:

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) 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,∞) B.

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

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

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?

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?

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

A. 0 # w # 45

B. 0 , w , 45

C. 0 # w # 3 45

D. 0 , w , 3 45

5. Reason abstractly. Why is the volume function of a prism or cylinder a third-degree equation? c. Approximate the coordinates of the maximum point of the function.

L esson 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.

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.

7. Which of the following is NOT a polynomial? A. f (x) 5 3x3 2 5x2 1 7x 2 11 1 B. f (x) 5 x2 1 5x3 2 8 2 C. f (x) 5 0.86x 2 6x4 1 3x2 2 5x3 1 9

c. Find the radius that yields the maximum value. d. Find the maximum volume of the tube. © 2015 College Board. All rights reserved.

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D. f (x) 5 23x5 1 2x3 2 8x 2 1 6 1

SpringBoard Algebra 2, Unit 3 Practice

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9. Use appropriate tools.

8. Examine the graph.

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

y 5

5

25

date

x

25

a. Describe the leading coefficient.

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

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

b. Describe the end behavior 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. c. Name any x- or y-intercept(s) of the function.

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

d. Name any relative maximum values and minimum values of the function. b. f (x) 5 23x3 1 2x

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

12. Determine whether the function is even, odd, or neither. Justify your answer. y 5

5

25

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

x

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.

25

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

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

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.

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) C.  (2, 7)

B. (2, 27) D. (7, 22) c. Compare the value of R(t) to the value of C(t) and P(t) for every t.

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. © 2015 College Board. All rights reserved.

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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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SpringBoard Algebra 2, Unit 3 Practice

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

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

date

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

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).

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

R(t) 5 51t3 2 892t2 1 5400t 1 200 C(t) 5 65t3 2 770t2 1 990t 1 9000

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?

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

Lesson 15-2 21. Find each sum or difference. Write your answer as a polynomial in standard form.

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

a. (5x4 1 7x3 1 2x2 1 25x 2 9) 1 (6x4 2 9x3 1 8x2 2 17x 1 5)

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

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

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.

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

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SpringBoard Algebra 2, Unit 3 Practice

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

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

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.

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

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

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

22. Find each product. Write your answer as a polynomial in standard form. a. (x 2 3)(x2 1 9x 2 2)

Lesson 15-3 26. Use long division to find each quotient. b. (x 2 5x 1 13)(3x 2 2)

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

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

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

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

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

3

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

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

A. 3x2 2 6 B. x6 2 8 C. 3x6 2 6x2 2 8 D. x3 2 6x2 1 12x 2 8

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SpringBoard Algebra 2, Unit 3 Practice

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27. Use synthetic division to find each quotient.

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. (x 1 5x 1 x 2 14x 1 3) 4 (x 1 3) 4

3

date

2

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

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

c. (x 1 2x 2 13x 1 10) 4 (x 1 5) 3

2

Lesson 16-1 31. Evaluate each combination. a. 12C8

d. (2x5 2 8x4 1 5x3 2 20x2 2 7x 1 28) 4 (x 2 4) b. 6C2

 20 c.    15 

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

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 18 d.    12

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37. Find the specified term of each expansion.

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.

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

33. Which row of numbers represents the fifth row of Pascal’s Triangle? b. the fifth term in (x 1 7)16

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

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

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?

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

Lesson 16-2 36. Express regularity in repeated reasoning. Find the coefficient of the specified term in each expansion.

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

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

b. the second term in (x 1 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

c. the fifth term in (x 1 6)10 40. Write and evaluate the expression  n  n 2 (r 2 1) r 2 1 b for n 5 5, r 5 3, a 5 2x,  r 2 1 a and b 5 4.

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

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SpringBoard Algebra 2, Unit 3 Practice

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43. Factor each sum or difference of cubes.

Lesson 17-1

a. x3 1 125

41. Factor each polynomial. a. x2 2 7x 1 12

b. x3 2 8 b. 3x2 1 x 2 10 c. 8x3 1 216

c. 3x4 1 2x2 2 5

d. 64x3 2 27

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

d. x2 1 5x 2 36

a. 25x4 2 169

b. x4 1 6x2 1 9

42. Factor by grouping. a. 2x3 2 6x2 1 5x 2 15

c. x6 2 10x3 1 25 b. 3x4 2 x3 1 6x 2 2

d. 4x10 2 81

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

c. x3 1 5x2 2 9x 2 45

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. x3 2 5x2 2 3x 1 15

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

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SpringBoard Algebra 2, Unit 3 Practice

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

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

a. f (x) 5 x3 1 4x b. n 5 4; zeros: 23, 2, 61

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

b. g(x) 5 x4 2 81

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

c. ℎ(x) 5 2x4 2 16x3 1 32x2

49. Which is the degree of the polynomial function 2 with the roots x 5 23, x 5 , x 5 i, and x 5 1 – i? 3 A. 4 B. 5 C. 6

D. 8

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

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

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

b. n 5 4; x 5 2i, x 5 1 1 3i 47. Attend to precision. Complete the following statement:

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

As a consequence of the Fundamental Theorem of Algebra,

1 1 d. n 5 4; x 5 i, x 5 6 2 2

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SpringBoard Algebra 2, Unit 3 Practice

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I.

Lesson 18-1

y 5

51. Match each equation to its graph. 1 a. f (x) 5 2 x 2 2 3 b. g(x) 5 x2 1 3

210

5

25

10

c. ℎ(x) 5 2x3 2 3x2 1 1 1 d. j(x) 5 x4 2 3x2 1 4 2 1 e. k(x) 5 2 x5 2 x4 2 x3 2 x2 2 x 2

25

210

II.

y 10

y

I II.

10

5 5

5

25

x 5

25

x

25 25

IV.

y 10 210

V.

5

22

1

21

2

x

y 15

10

25

5

210 25

© 2015 College Board. All rights reserved.

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10

5

x

SpringBoard Algebra 2, Unit 3 Practice

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52. For polynomials, relative extrema occur

date

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

A. where the graph crosses the y-axis.

a. ℎ(x) 5 x3 2 x2 1 3x 1 5

B. at the ends of the polynomial function. C. where the graph crosses the x-axis. D. between the zeros of the polynomial function.

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

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

59. Given k(x) 5 x3 2 2x2 2 5x 1 6: a. Find the real zeros of k(x).

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.

b. What is the y-intercept?

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

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

b. Find all the intercepts of the function. c. Find the relative extrema of the function.

d. Graph k(x) by hand.

Lesson 18-2

y

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:

x

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? © 2015 College Board. All rights reserved.

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SpringBoard Algebra 2, Unit 3 Practice

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64. Solve each inequality.

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

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

I. m(23) 5 0 II. x 1 3 is a factor of m(x). I II. m(x) also has an x-intercept at (3, 0). b. x3 2 5x2 2 2x 1 24 # 0

A. I only

B.  II only

C. III only

D.  II and III only

Lesson 18-3

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

61. Which representation below is a quadratic function that has zeros at x 5 2 and x 5 4? Explain.

f(x)

A. ℎ(x) 5 x2 2 6x 1 8 B.

  0 24

x y

24

1 22

2 0

3 2

4 4

20 16 12

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

8 4

q(x) 10 28

4

24

8

x

24 28

5

212

5

25



x

216

A. 2∞ # x $ ∞ B. 25 # x #21, and x $ 2 C. 25 # x #21 and x # 2

25

D. 2∞ # x #25, and 21 # x # 2

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.

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SpringBoard Algebra 2, Unit 3 Practice