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Oct. 23 • A23 Due • Friday - Test on Chapter 3 • A24: Chapter 3 Practice Test (Due Friday) • Key is on bulletin board • For last problem, see hint! • Today - Start Chapter 5 on Quadratics • A25: p. 245 #1, 5, 9, 11, 17, 29, 33, 37 (DUE TUESDAY) Warm up – October 23 notes Multiply and Simplify: 1) y = x(1- x) - (1- x2 ) Which of the above is a: • Linear Function • Quadratic Function 2) y = (x + 3)(x -1) 5-1: Modeling Quadratic Functions Define: • Linear Function – • Quadratic Function – 5-1: Modeling Quadratic Functions Define: • Linear Function – A function whose graph is a line Slope-intercept form y = mx + b Standard form Ax + By = C • Quadratic Function – A function whose graph is a parabola Standard form y = ax2 + bx + c Quadratic Term Linear Term Constant Term Key elements of a parabola • Line (or axis) of Symmetry • the line that divides a parabola into two parts that are mirror images • Vertex of Parabola • the point on the line of symmetry. • y-value represents the max or min of the function. Identify key elements of this parabola • Axis of symmetry P (0,6) • Vertex • Corresponding (mirror) points to: • P (0,6) => P’ ( Q(1,0) • Q (1,0) => Q’ ( ) ) Finding Quadratic function given 3 points Lines • We need 2 points to find the equation of a line • (or 1 point and slope) Quadratics • We need 3 points to find the equation of a parabola y = mx + b y = ax2 + bx + c y = 2x - 4 y = -3x2 + 5x -13 To find the Quadratic Function, use the 2 y = ax + bx + c Standard Form of Quadratics Example: Find the Quadratic, given the points (1, 0) , (2, -3) and (3, -10) What do we know / not know? y = ax + bx + c 2 • Steps: • 1) Substitute the three points into standard form to get 3 equations • 2) Solve the system of equations to get the values for a, b, and c • 3) Write the standard form of the equation substituting in the values of a, b, and c. y = -2x + 3x -1 2 Oct 29 • Due: A25 (p. 245) and A26 (worksheet) • Tests are graded, but 3 people still need to take it • HW Review • 5-2 Properties of Parabolas • CW: Quadratic Functions Worksheet • A27: p. 252 #1-21 EOO, & #18 5-2 Properties of Quadratics Graphing function y = ax2+c • What is the max or min y value for? y= -0.5x2 + 2 • How can we find other points? When no linear term, the vertex is (0,c) • If y = ax2 + c, • Then the vertex is (0, c) & the axis of symmetry is x = 0 • Check: 1. Sketch graph of y = 2x2 – 4 2. Sketch graph of y = 5 – 3x2 3. What are the coordinates of the vertex of the graph of a function in the form y = ax2 ? If y = ax2 + bx + c • If a is positive, it opens __________ • If a is negative, it opens ___________ b • The axis of symmetry is the line x = 2a • The x-coordinate of the vertex is __________ • The y-coordinate of the vertex is found by____________ • The y-intercept is ( ) Steps to Graphing a quadratic 1. Find and graph the 2. 3. 4. 5. axis of symmetry Find and graph the vertex Find and graph the yintercept and its reflection Pick another value for x, evaluate for y and graph that point and its reflection Sketch the curve y = -x2 + 4x + 2 y= - 4x2 + 8x - 5 Oct 31 • Due: A26 (worksheet) and A27 • Warm up • HW Review • 5-2 Properties of Parabolas / 5-3 Transforming Parabolas • CW: Quadratic Functions Worksheet • A28: p. 252 #26-32, 37-39, 55; p. 259 #1-37 EOO Warm up Quiz – in your Notes • 1) Find a Quadratic function to model the points below (-2, -18) (0, -4) (2, -14) • 2) Sketch the Graph of the following function. Label the vertex and axis of symmetry. y = -2x - 4x + 2 2 HW Questions? 5-2 Using Vertex to find Min/Max • You can find the min or max of a quadratic function by finding the vertex. • A collage frame must be a rectangle with a perimeter of 25 cm. What dimensions give the maximum area? 5-3 Vertex form of Parabolas • We have been studying • Standard Form: y = ax + bx + c 2 • Today you will be learning vertex form y = a(x - h) + k 2 Complete empty columns Standard Form y = ax2 + bx + c b x=2a Vertex form y = a(x - h)2 + k y = x - 4x + 4 y = (x - 2)2 y = x + 6x + 8 y = (x + 3)2 -1 y = -3x2 -12x - 8 y = -3(x + 2)2 + 4 2 2 a Vertex is (h, k) a is same as in Standard Form h Complete empty columns Standard Form y = ax2 + bx + c b x=2a Vertex form y = a(x - h)2 + k y = x - 4x + 4 y = (x - 2)2 y = x + 6x + 8 y = (x + 3)2 -1 y = -3x2 -12x - 8 y = -3(x + 2)2 + 4 2 2 a h Attributes of •a •h •k y = a(x - h)2 + k Using Vertex form to graph 1. Graph the vertex & y = -3(x + 2)2 + 4 label the axis of symmetry 2. Find another point and its reflection. Graph these points 3. Sketch the curve Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/ Graph the function y = 2(x +1) - 4 2 Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/ Writing the equation of a parabola • Vertex (3, 4) • Point (5, -4) 1. Write vertex form 2. Substitute the vertex for h, k 3. Plug in 2nd point and solve for a. 4. Write equation using vertex form You Try! • Vertex (-1, 0) • Point (-2, 2) Converting to Vertex form 1. Find the x- coordinate of the vertex 2. Find the y- coordinate by plugging in x 3. a is given 4. Substitute vertex for (h, k) and a from initial equation y = x2 - 4x + 4 You try! y = -3x +12x + 5 2 Review • Page 262, #94 Nov 4 • Due: A28: p. 252 #26-32, 37-39, 55; p. 259 #1-37 EOO • Warm up • 5-4 Factoring Quadratics • Quiz • A29: p. 267 #1-49 EOO Warm up Quiz – in your Notes • 1) Rewrite the equation in vertex form. Identify the vertex and the axis of symmetry. Sketch the graph of the function. y = -3x - 6x - 8 2 Multiply using Generic Rectangles (aka Area Models) (x -1)(3x +12) 5-4: Factoring Quadratics • If possible, always factor out GCF first! • GCF: Greatest Common Factor 4x + 20x -12 2 9n - 24n 2 9x + 3x -18 2 4w + 2w 2 When a = 1 • When the coefficient of the x2 term is 1, use diamonds When a is not 1: Rectangles / Diamonds Try: 2x +11x +12 2 3x +12x -15 2 Quiz • Individual. You can use your notes. • After quiz, work on HW • A29: p. 267 #1-49 EOO Nov 6 • Due: A29: p. 267 #1-49 EOO • Finish 5-4 Factoring Quadratics • Start 5-5 Quadratic Equations • Finish Quiz • A30: p. 270 Odds, (Due Friday) • A31: p. 274 #1-19 all (TBD – might change on Friday!) Warm up – In your Notes • Factor: 2x +10x - 28 2 4x -1 2 Other methods for factoring exist • Your text has a method similar to box/diamond, (see examples on pages 264-265) • There is also the AC method, which works, but I’m not a huge fan • http://www.regentsprep.org/regents/math/algtrig/ATV1/Ltri3.htm • And many others, e.g. Guess and Check • What works best is…… what works best for you! Two Special Expressions that are good to know: • Perfect Squares a2 + 2ab + b2 = (a + b)2 a - 2ab + b = (a - b) 2 2 2 If you can find the square root of the first and last terms, AND the middle term is twice the product of the roots • Difference of two Squares a - b = (a + b)(a - b) 2 2 If you can find the square root of the first and last terms, AND there is no middle term AND the sign in between is - Examples 4x + 20x + 25 2 4x - 20x + 25 2 16x - 25 2 16x + 25 2 Nov 8 • Due today: A30: p. 270 Odds • Warm up • Start 5-5 Quadratic Equations • A31: p. 274 #1-19 odd, 37-53 EOO Warm up – in your notes! • Factor 16x + 40x + 25 2 36x - 24x + 4 2 49x -1 2 16x + 64 2 5-5 Quadratic Equations • What values of x would make this equation true? ( x + 3)( x - 7) = 0 • What point(s) would that be? • Where would that be on a graph? Ways to solve quadratics • Zero Product Property • Square Roots • Graphing Calculators (Graph/Table) Zero-Product Property Used when we have a quadratic equation that can be factored. if ab = 0, then a = 0 or b = 0 If two things are being multiplied together and the result is zero, then one of those has to equal Zero. Example Steps 1. Solve for zero. (Write equation on standard form) 2. Factor 3. Set each factor = 0 and solve. 4. Check! 2x -11x = -15 2 You Try! x + 7x = 18 2 2x + 4x = 6 2 16x = 8x 2 Solving by square roots 5x -180 = 0 2 • This is useful when we have no linear term (no bx term) Steps 1. ax2 term on left, number right 2. Solve for x2 (divide by a) 3. Find square roots You Try! 4x - 25 = 0 2 3x = 24 2 1 x - =0 4 2 Solving with a Graphing Calculator • Enter the following function into Y= y = x - 5x + 2 2 Classwork – in pairs • Put both names on one page (OR turn in 2 pages, your choice) • Solve by Graphing/Table: p. 274 #23, 27, 31 • Solve any way: p. 275 #35 • A31: p. 274 #1-19 odd, 37-53 EOO Nov 13 • Due today: A31: p. 274 #1-19 odd, 37-53 EOO • Complex Numbers • Quadratic Equation Activity • A32: p. 282 #3-43 EOO, 49, 52a, 55 • p. 285 #1-9 odd • A35: Ch 5 Review (DUE NEXT THURS 11/21) • CHAPTER 5 TEST NEXT THURS `11/21 Nov 13 • Due today: A31: p. 274 #1-19 odd, 37-53 EOO • Complex Numbers • Quadratic Equation Activity • A32: p. 282 #3-43 EOO, 49, 52a, 55 • p. 285 #1-9 odd • A35: Ch 5 Review (DUE NEXT THURS 11/21) • CHAPTER 5 TEST NEXT THURS `11/21 Nov 15 • Due today: A32: p. 282 #3-43 EOO, 49, 52a, 55 • p. 285 #1-9 odd • Complex Numbers (Division and solving) • Completing the square • A33: p. 289 #1-33 EOO (do #25, 29 in class) • A35: Ch 5 Review (DUE NEXT THURS 11/21) • CHAPTER 5 TEST NEXT THURS 11/21 Warm up: • Sketch the graph of this function • Label key information Quadratics…what have you learned? • 5-1: vertex and axis of symmetry; how to find standard • • • • • • • • form 5-2: standard form and how to find vertex, a.o.s; which way will the parabola open, min/max, y-intercept? 5-3: vertex form; how to find vertex 5-4: factoring; what does that mean in the graph? <simplifying square roots> 5-5: solving by factoring; solving by square roots (table/graph too) 5-6: complex numbers 5-7: solving by completing the square; vertex form 5-8: solving using quadratic formula; discriminant Complex Numbers • Division Completing the square • OBJECTIVE: You’ll use completing the square to solve quadratic equations. • The process of completing the square is a really useful method that can help solve quadratic equations. • The best way to show how useful completing the square can be is with an example… EXAMPLE: Solve x2 – 10x + 21 = 0 by completing the square. • Step 1: Move the constant (c) to the right by subtracting from both sides • Step 2: Figure out what you need to complete the square on the left side (to make a perfect square) • Step 3: ADD THIS NUMBER TO BOTH SIDES to preserve the equality • Step 4: Factor the left and simplify the right • Step 5: Now solve by taking the square root of both sides You Try! x2 – 6x + 5 = 0 x2 + 4x – 5 = 0 What if there is an (a) term? • Step 0: Divide everything by a • Step 1: Move the constant (c) to the right by subtracting from both sides • Step 2: Take half of b and square it • Step 3: ADD TO BOTH SIDES • Step 4: Factor the left and simplify the right • Step 5: Solve • Example: 4x2 – 9x + 2 = 0 You Try! 3a2 + 12a – 15 = 0 y = x2 + 6x + 2 Vertex form using CTS • Before you take square root to solve, just move number back! Nov 19 Due today: A33: p. 289 #1-33 EOO (do #25, 29 in class) • Quadratic Formula / Discriminant • A34: p. 297 #1-49 EOO, 57-59 • A35: Ch 5 Review (due Monday after Thanksgiving) Warm up • Rewrite the following equation in Vertex Form. • Then sketch the graph. y = 2x - 8x +1 2 Deriving the Quadratic Formula -b ± b2 - 4ac x= 2a ax2 + bx + c = 0 The quadratic formula gives roots to our equations • Use quadratic formula to solve 3x2 - 5x = 2 We can get complex solutions 2x = -6x - 7 2 You Try~! 3x - x = 4 2 -2x2 = 4x + 3 How can we tell the quadratic has real roots or not? -b ± b2 - 4ac x= 2a The Discriminant • If b - 4ac > 0 2 b - 4ac = 0 2 b - 4ac < 0 2 Then: Determine the type and # of solutions x2 + 6x + 8 = 0 x + 6x + 9 = 0 2 x2 + 6x +10 = 0 Determining how to solve quadratics If the Discriminant is…. a Positive Square Number Factor Table/ Graph Quadratic Formula Complete the Square ✔ ✔ ✔ ✔ (approximate) ✔ ✔ ✔ ✔ ✔ ✔ ✔ a Positive non-square number Zero a Negative ✔ • https://www.youtube.com/watch?v=z6hCu0EPs-o • https://www.youtube.com/watch?v=2lbABbfU6Zc Nov 21 Objective: Today you will assess your readiness for Chapter 5 test! Method: You will take a partner quiz – Great chance to LEARN, debate, figure out what you need to practice! Due today: • A34: p. 297 #1-49 EOO, 5759 • PARTNER QUIZ • A35: Ch 5 Review (due Monday after Thanksgiving) • Ch 5 TEST on Wednesday after Break!