Algebra 2 Distance Learning Packet Teacher: ______________________ Teacher’s Email: _____________________ School: ______________________ Virtual Office Hours: 9:00 a.m.- 11:00 a.m. & 1:00 p.m.- 3:00 p.m. Conference Call Dial-in Number: ______________________ Dial-in Access Code: ___________ Online Meeting URL: __________________________________ Online Meeting ID: ______________________________ April 13 April 14 April 15 April 16 April 17 Standard: A2.AAPR.3 Standard: A2.AAPR.3 Standard: A2.AAPR.3 Standard: A2.AAPR.3 Standard: A2.AAPR.3 Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Complete 4-4 Study Guide and 4-4 Complete 4-5 Study Guide and Graph your assigned polynomial Determine the characteristics of Write a paragraph describing the Word Problem Practice Intervention & Skills Practice function. your assigned polynomial characteristics of your assigned function. polynomial function. Extension Activities Extension Activities Extension Activities Extension Activities Extension Activities Khan Academy Polynomial Graphs Khan Academy Polynomial Graphs Khan Academy Polynomial Graphs Khan Academy Polynomial Graphs Unit – Zeros of Polynomials Unit – + & - Intervals Unit – End Behavior Unit – Put it all together https://www.khanacademy.org/ https://www.khanacademy.org/ https://www.khanacademy.org/ https://www.khanacademy.org/ math/algebra2/ math/algebra2/ math/algebra2/ math/algebra2/ x2ec2f6f830c9fb89:poly-graphs x2ec2f6f830c9fb89:poly-graphs x2ec2f6f830c9fb89:poly-graphs x2ec2f6f830c9fb89:poly-graphs April 20 April 21 April 22 April 23 April 24 Standard: A2.AAPR.1 Standard: A2.AAPR.1 Standard: A2.AAPR.1 Standard: A2.AAPR.1 Standard: A2.AAPR.1 Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Complete 5-1 Study Guide Complete 5-3 Study Guide Operations with Functions – April Standardized Test Practice #1-6 Standardized Test Practice #7-12 22 #1-10 Extension Activities Extension Activities Extension Activities Extension Activities Extension Activities Composing Functions Verifying Inverse Functions https://www.khanacademy.org/ https://www.khanacademy.org/ math/precalculus/ math/precalculus/ x9e81a4f98389efdf:composite x9e81a4f98389efdf:composite SHG JMW Algebra 2 Distance Learning Packet April 27 April 28 April 29 April 30 May 1 Standard: A2.ASE.2 Standard: A2.AREI.2 Standard: A2.AREI.2 Standard: A2.AREI.2 Standard: A2.NCNS.1 Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Learning Tasks: Factor – April 27 #1-8 Solving Equations – April 28 #1-10 Radicals – April 29 #1-10 Rational Exponents – April 30 #1-10 Complex Numbers – May 1 #1-10 Extension Activities Extension Activities Extension Activities Extension Activities Extension Activities Factoring Practice Solving Practice Simplifying Roots Tutorial Fractional Exponents Practice Complex Numbers Tutorial https://www.thatquiz.org/tq- https://www.thatquiz.org/tq-0/?- https://schoolyourself.org https://www.khanacademy.org/math/ https://schoolyourself.org/learn/ 0/?-jh00-l8-mpnv600-p0 j102-l6-mpnv600-p0 /learn/algebra/simplifying-roots algebra2/x2ec2f6f830c9fb89:exp/ precalculus/complex-numbers x2ec2f6f830c9fb89:rational- exp/e/exponents_3 SHG NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 4-4 Study Guide and Intervention Graphing Polynomial Functions Graphs of Polynomial Functions If the degree is even and the leading coefficient is positive, then f(x) → +∞ as x → –∞ f(x) → +∞ as x → +∞ If the degree is even and the leading coefficient is negative, then End Behavior of f(x) → –∞ as x → –∞ f(x) → –∞ as x → +∞ Polynomial Functions If the degree is odd and the leading coefficient is positive, then f(x) → –∞ as x → –∞ f(x) → +∞ as x → +∞ If the degree is odd and the leading coefficient is negative, then f(x) → +∞ as x → –∞ f(x) → –∞ as x → +∞ The maximum number of zeros of a polynomial function is equal to the degree of the polynomial. Real Zeros of a A zero of a function is a point at which the graph intersects the x-axis. Polynomial On a graph, count the number of real zeros of the function by counting the number of times Function the graph crosses or touches the x–axis. Example: Determine whether the graph represents an odd-degree polynomial or an even-degree polynomial. Then state the number of real zeros. As x → –∞, f(x) → –∞ and as x → +∞, f(x) → +∞, so it is an odd-degree polynomial function. The graph intersects the x–axis at 1 point, so the function has 1 real zero. Exercises For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeroes. 1. 2. 3. Chapter 4 23 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 4-4 Word Problem Practice Graphing Polynomial Functions 1. MANUFACTURING A metal sheet is curved 4. DRILLING The volume of a drill bit in cubic according to the shape of the graph of f(x) = 푥4 – 9푥2. millimeters can be estimated by the formula for a 1 What is the degree of this polynomial? cone, V = 휋ℎ푟2, where h is the height in millimeters 3 of the bit and r is its radius in millimeters. 3 Substituting √ r for h, the volume of the drill bit is 3 3 estimated as √ 휋푟3. Graph the function of drill bit 2. GRAPHS Kendra graphed the polynomial f(x) shown 9 below. volume. Describe the end behavior, degree, and sign of the leading coefficient. From this graph, describe the end behavior, degree, and sign of the leading coefficient. 5. TRIANGLES Dylan drew n dots on a piece of paper making sure that no line contained 3 of the dots. The number of triangles that can be made using the dots as 1 3 2 vertices is equal to f(n) = (푛 – 3푛 + 2n). 6 3. PENTAGONAL NUMBERS The nth pentagonal 푛(3푛 − 1) number is given by the expression . 2 What is the degree of this polynomial? What is the seventh pentagonal number? a. What is the degree of f ? b. If Dylan drew 15 dots, how many triangles can be made? Chapter 4 27 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 4-5 Study Guide and Intervention Analyzing Graphs of Polynomial Functions Location Principle Suppose y = f(x) represents a polynomial function and a and b are two numbers such Location Principle that f(a) < 0 and f(b) > 0. Then the function has at least one real zero between a and b. Example: Determine consecutive integer values of x between which each real zero of f(x) = 2풙ퟒ – 풙ퟑ – 5 is located. Then draw the graph. Make a table of values. Look at the values of f(x) to locate the zeros. Then use the points to sketch a graph of the function. x f(x) The changes in sign indicate that there are zeros between x = –2 and x = –1 and between x = 1 and x = 2. –2 35 –1 –2 0 –5 1 –4 2 19 4-5 Study Guide and Intervention (continued) Analyzing Graphs of Polynomial Functions Maximum and Minimum Points A quadratic function has either a maximum or a minimum point on its graph. For higher degree polynomial functions, you can find turning points, which represent relative maximum or relative minimum points. Example: Graph f(x) = 풙ퟑ + 6풙ퟐ – 3. Estimate the x-coordinates at which the relative maxima and minima occur. Make a table of values and graph the function. x f(x) A relative maximum occurs at x = –4 and a relative minimum occurs at x = 0. –5 22 –4 29 ← indicates a relative maximum –3 24 –2 13 –1 2 ← zero between x = –1, x = 0 ←indicates a relative minimum 0 –3 1 4 2 29 Chapter 4 29 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 4-5 Skills Practice Analyzing Graphs of Polynomial Functions Complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive values of x between which each real zero is located. c. Estimate the x-coordinates at which the relative maxima and minima occur. 1. f(x) = 푥3 – 3푥2 + 1 2. f(x) = 푥3 – 3x + 1 x f(x) x f(x) –2 –3 –1 –2 0 –1 1 0 2 1 3 2 4 3 3 2 3. f(x) = 2푥 + 9푥 +12x + 2 4. f(x) = 2푥3 – 3푥2 + 2 x f(x) x f(x) –3 –1 –2 0 –1 1 0 2 1 3 4 2 5. f(x) = 푥 – 2푥 – 2 6. f(x) = 0.5푥4 – 4푥2 + 4 x f(x) x f(x) –3 –3 –2 –2 –1 –1 0 0 1 1 2 2 3 3 Chapter 4 31 Glencoe Algebra 2 Week 3 Polynomial Project Directions: 1. Choose your assigned function based on the first letter of your last name: 2. Fill out the “Characteristics of a Polynomial Function” graphic organizer. • A-B: 푦 = 2푥3 − 4푥 + 6 • C-D: 푦 = 2푥4 − 4푥2 − 2 3. Write a paragraph detailing all of the key characteristics for the graph to • E-F: 푦 = 2푥4 − 4푥2 − 2푥 include: • G-H: 푦 = −2푥4 + 3푥2 + 1 • type of polynomial function • I-J: 푦 = −2푥3 + 3푥2 + 푥 • domain • K-L: 푦 = −2푥3 + 3푥2 + 푥 − 4 • range • M-N: 푦 = −2푥3 + 푥 − 4 • end behavior • O-Q: 푦 = −2푥2 + 6푥 + 7 • intervals of increase and/or decrease • R-S: 푦 = 3푥2 + 6푥 − 1 • turning points (including whether it’s a relative maximum or relative • T-V: 푦 = 3푥5 + 2푥3 + 푥2 − 푥 − 5 minimum) 1 • W-Z: 푦 = − 푥5 + 3푥2 + 푥 − 5 • zeros and their multiplicity 2 • the effect of the multiplicity on the graph. NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 5-1 Study Guide and Intervention Operations with Functions Perform Operations with Functions You can use these rules to perform operations with functions. Sum (f + g)(x) = f (x) + g(x) Difference (f – g)(x) = f (x) – g(x) Product (f ⋅ g)(x) = f (x) ⋅ g(x) 푓 푓(푥) Quotient ( )(x) = , g(x) ≠ 0 푔 푔(푥) 풇 Example: Find (f + g)(x), (f – g)(x), (f ⋅ g)(x), and ( ) (x) for f(x) = 풙ퟐ + 3x – 4 and g(x) = 3x – 2.