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Warm Up 3.20

MM2A3 c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema , intervals of increase and decrease, and rates of change. Warm Up 3.20. Find the following. Domain: Range : Vertex: x-intercepts:

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Warm Up 3.20

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  1. MM2A3c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. Warm Up 3.20 Find the following. • Domain: • Range: • Vertex: • x-intercepts: • y-intercepts: • Interval of Increase: • Interval of Decrease: • Rate of Change of the given points: 10 5 -5 5

  2. MM2A3c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. Warm Up 3.20 Find the following. • Domain: • Range: • Vertex: • x-intercepts: • y-intercepts: • Interval of Increase: • Interval of Decrease: • Rate of Change of the given points: (-∞, +∞) 10 5 (-∞, 13] (-2, 13) X = -6, 2 y = 10 (-∞, -2) -5 4 (-2, +∞) (-5, 5) and (0, 10) y1 – y2 _ 5 – 10 _ -5 x1 – x2 – -5 – 0 – -5 = = = 1

  3. Transformation Form Recall: a(x-h)n + k = 0 Example: 1(x+3)2 + 0 = 0 We have learned a is the stretch/shrink a = 1 (no stretch or shrink) We have learned h is the horizontal shift h = 3 (moves left) We have learned k is the vertical shift k = 0 (no vertical shift)

  4. Standard Form Recall: ax2+bx+c = 0 Example: x2 + 6x + 9 = 0 (-2)2+6(-2)+9 = 4 – 12 + 9 = -8 + 9 = 1 X Y 0 -2 -4 1 -3 1 We have learned how to factor: (x +3)(x+3) And solve: x = -3 (-4)2+6(-4)+9 = 16 – 24 + 9 = -8 + 9 = 1 Let’s graph it! This is your x - intercept

  5. Standard Form and Transformation (Vertex) Form Examples: x2 + 6x + 9 = 0 OR 1(x+3)2 + 0 = 0 SAME graph 

  6. Unit 5: Quadratics and Complex Numbers MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k.

  7. Standards Today Unit 5 • MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k. • MM2N1. Students will represent and operate with complex numbers. • MM2A4. Students will solve quadratic equations and inequalities in one variable. a. Convert between standard and vertex form. b. Graph quadratic functions as transformations of the function f(x) = x2. c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. d. Explore arithmetic series and various ways of computing their sums. e. Explore sequences of partial sums of arithmetic series as examples of quadratic functions.

  8. Essential Question What is vertex form? What is a vertex? Can I convert from vertex form to standard form?

  9. y = a(x-h)n + k This equation has a new name… VERTEX FORM

  10. I. Quadratics • Vertex form: a(x-h)2 + k = 0 • Vertex– the maximum or minimum point of a quadratic function; vertex point is (h, k) (quadratics are also called a parabola) • Axis of Symmetry - the vertical line through the vertex of a quadratic (parabola); x = h h k

  11. Vertex- . ….. • The lowest or highest point of a parabola. Vertex Axis of symmetry- • The vertical line through the vertex of the parabola. Axis of Symmetry

  12. Graph Example Where is the Vertex? Axisof Symmetry? How do you write this? Equation: x = h (from vertex) OR x = - This is Standard Form b 2a -4 -4 2(2) 4 = - = - = - (-1) = 1

  13. Vertex Form Remember: y = a(x-h)2 + k vertex (h, k) Example: y = (x+2)2 – 1 vertex: (-2, -1) Vertex form  Standard form (x+2)2 – 1 = (x+2)(x+2) – 1 = x2 + 2x + 2x + 4 – 1 = x2 + 4x + 3 What is the vertex? Recall: y = ax2 + bx+ c Recall: when h is positive  x-h when h is negative  x – (-h) x + h *Always switch the sign of h* Rewrite using Binomial Theorem Use FOIL to multiply then gather like terms We graphed this in class.  Standard Form

  14. You try: a) y = (x – 1)2 + 3 b) y = (x+3)2 – 2 (x – 1)(x – 1) + 3 = x2 – 2x + 1 + 3 = x2 – 2x + 4 Expand Gather Like terms (x + 3)(x + 3) – 2 = x2 + 6x + 9 – 2 = x2 + 6x + 7 Expand Gather Like terms

  15. Homework Example Example 2: Vertex to Standard y = 2(x + 1)2 + 2 • Vertex • Transformations • Standard Form (-1, 2) Shrinks by 2, moves to the left 1 and shifts up 2 2(x + 1)2 + 2 =2(x+1)(x+1) + 2 =2(x2+2x+1) + 2 =2x2+ 4x + 2 + 2 =2x2 + 4x + 4 Expand Use FOIL Distribute Gather Like terms  Standard form

  16. Homework a) Write vertex. b) Write in standard form. c) Write all transformations in a complete sentence. • y = 2(x + 4)2 – 6 • y = (x – 2)2 + 1 • y = (x + 3)2 – 5 • y = ½(x – 7)2 – 1 • y = 3(x+ 6)2 + 6 4th White ONLY 6. y = (x + 2)2 – 3 7. y = ¼ (x + 8)2 + 1 8. y = (x + 1)2 – 1 9. y = (x – 5)2 + 4 10. y = 3(x – 9)2 – 18

  17. Next Class Standard  Vertex 2x2 + 4x + 4 Step 1: Group x terms (2x2 + 4x)+ 4 =2(x2 + 2x) + 4 Step 2: Take ½ of b and square (2) it. Add this inside () and subtract from the outside. 2(x2 + 2x + 1)+ 4 – 2(1) Step 3: Factor and Simplify *a should always equal 1* Find GCF of coefficients so a = 1 2(x+1)(x+1)+ 4 – 2 = 2(x+1)2 + 2 ( )2 b 2 Vertex Form

  18. Recall :Essential Question What is vertex form? What is a vertex? Can I convert from vertex form to standard form? y = a(x-h)2+ k the maximum or minimum point of a quadratic function

  19. Unit 5: Quadratics and Complex NumbersDay 2 MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k.

  20. Standards • MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k. a. Convert between standard and vertex form. b. Graph quadratic functions as transformations of the function f(x) = x2. c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change.

  21. Essential Question Can I convert form standard form to vertex form? Can I identify the vertex? Can I identify the axis of symmetry?

  22. Quadratics cont. y = 2x2 – 4x – 1 How do you get it to… y = 2(x – 1)2 – 3?? Standard Form Vertex Form Completing the Square

  23. ) ) ( ( b –10 to each side. Add Add to each side. 2 2 25 (–5) 2 = = 2 2 ANSWER The vertex form of the function is y = (x – 5)2 – 3 . The vertex is (5, –3). The axis of symmetry is x = 5. Write a quadratic function in vertex form Write y = x2 – 10x + 22 in vertex form. Then identify the vertex. y = x2 – 10x + 22 Write original function. y + ?= (x2–10x + ?) + 22 Prepare to complete the square. b y + 25= (x2– 10x + 25) + 22 y + 25 = (x – 5)2 + 22 Factor x2 – 10x + 25 and write it asbinomial squared. -25 -25 y = (x – 5)2– 3 Solve for y.

  24. Add something in to make a perfect square trinomial Subtract the same amount to keep it even. Now create a binomial squared This gives us the ordered pair (h,k) Vertex Form • Changing to vertex form ( ) ) ( 2 b 2 ) ( 2 b 2 b 2

  25. for Examples 6 and 7 GUIDED PRACTICE Write the quadratic function in vertex form. Then identify the vertex and axis of symmetry. 13. y = x2 – 8x + 17 y = (x – 4)2+ 1; (4, 1) ; x = 4. ANSWER 14. y = x2 + 6x + 3 y = (x + 3)2– 6 ; (–3, –6) ; x = -3 ANSWER 15. f(x) = x2 – 4x – 4 ANSWER y = (x – 2)2– 8 ; (2 , –8) ; x = 2

  26. for Examples 6 and 7 GUIDED PRACTICE 16. What if ? In example 7, suppose the height of the baseball is given by y = – 16t2 + 80t + 2. Find the maximum height of the baseball. Find maximum point -> find vertex  find y value (vertical point) ANSWER 102feet.

  27. Recall: Essential Question Can I identify the vertex? Can I identify the axis of symmetry? Can I convert form standard form to vertex form?

  28. Additional Slides The following slides are for additional practice. You can “unhide” the slides to view in the slideshow Go to slideshow and click on the hidden slides then unclick “hide” This must be done individually for EACH hidden slide

  29. Unit 5: Quadratics and Complex NumbersDay 3 MM2A4. Students will solve quadratic equations and inequalities in one variable.

  30. Standards • MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k. a. Convert between standard and vertex form. b. Graph quadratic functions as . transformations of the function f(x) = x2. c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. Graph quadratic functions as transformations of the function f(x) = x2.

  31. Essential Question How do I graph using vertex form? How do I graph using standard form?

  32. Graph from vertex Recall: y = a(x – h)2 + k Vertex: (h, k) Axis of Symmetry: x = h Example: y = -2(x + 1)2 + 3 Step 1: Find vertex Step 2: Pick 2 x points (1 before and 1 after vertex) Step 3: Solve for those two points 4: Graph 3 points and connect points (-1, 3) -1 3 X Y 0 -2 1 1 Solve for x = -2: y = -2(-2+ 1)2 + 3 y = -2(-1)2+ 3 y = -2(1) + 3 = -2 + 3 = 1 Solve for x = 0: y = -2(0 + 1)2 + 3 y = -2(1)2 + 3 y = -2(1) + 3 = -2 + 3 = 1 Let’s GRAPH it! 

  33. Graph from vertex Example: y = -2(x + 1)2 + 3 X Y 0 -1 -2 1 3 1 Great Job!

  34. You try: • y = (x – 6)2 + 3 2) y = -½(x + 2)2 – 1

  35. Graph from Standard Recall: y = x2 + bx + c Axis of Symmetry: - Vertex: plug in axis of symmetry (h) and solve for y (which gives you k) b 2a Example: y = x2 – 4x – 3 Step 1: Find axis of symmetry - = - = - = Step 2: Plug in axis value to find vertex Step 3: Find 2 points Step 4: Solve for those two points Step 5: Graph 3 points and connect points 1 X Y b 2a -4 2(1) -4 2 - (-2) = 2 1 3 -6 Find vetex, x = 2: y = (2)2 – 4 (2) – 3 y = 4– 8– 3 y = -4 – 3 = -7 -6 Vertex: ( 2, -7) 2-7 Solve for x = 1: y = (1)2– 4 (1) – 3 y = 1– 4– 3 y = -6 Solve for 3: y = (3)2– 4 (3) – 3 y = 9– 12– 3 y = -6 (1 before and 1 after vertex) Let’s GRAPH it! 

  36. Graph from standard Example: y = x2 – 4x – 3 X Y 1 2 3 -6 -7 -6 Great Job!

  37. You try: • y = x2 + 8x + 16 2) y = -2x2 + 2x – 3

  38. Recall: Essential Question How do I graph using vertex form? How do I graph using standard form? Start with the vertex Find two additional points Start with the axis of symmetry Find the vertex Find two additional points

  39. Additional Slides The following slides are for additional practice. You can “unhide” the slides to view in the slideshow Go to slideshow and click on the hidden slides then unclick “hide” This must be done individually for EACH hidden slide

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