Quadratics and their Properties

Transcription

1 Algebra 2 Quadratics and their Properties Name: Ms. Williams/Algebra 2 Pd: 1

2 Table of Contents Day 1: COMPLETING THE SQUARE AND SHIFTING PARABOLAS SWBAT: Write a quadratic from standard form to vertex form. Pgs. #3-7 HW: pg #8-9 in packet Day 2: More SHIFTING PARABOLAS SWBAT: Write a quadratic from standard form to vertex form. Pgs. #12-15 Hw: pg #12-15 in packet Day 3: Writing equations of a parabola using a directrix and focus. SWABT: Write an equation of a parabola using a directrix and focus. Pgs. # HW: pg #26-27 Day 4: More with writing equations of a parabola using a directrix and focus. SWABT: Write an equation of a parabola using a directrix and focus. Pgs. # HW: Page 35 # s odd and Pages all examples Addition Questions: Set A Page 40 Set B Page 42 2

3 Day 1 - COMPLETING THE SQUARE AND SHIFTING PARABOLAS SWBAT: Write a quadratic from standard form to vertex form. Warm - Up: Determine the value that would make each of the following a perfect square. a) x 2 + 6x + is a perfect square trinomial because it is = ( ) 2 b) x 2 20x + is a perfect square trinomial because it is = ( ) 2 What is the magic number that completes the square? Parabolas, and graphs more generally, can be moved horizontally and vertically by simple manipulations of their equations. This is known as shifting or translating a graph. You worked with this extensively in Common Core Algebra I. The first exercise will review how to use a method known as completing the square to identify shifts and the turning point of a parabola. Exercise #1: The function equation is 2 y x x y 2 x is shown already graphed on the grid below. Consider the quadratic whose (a) Using the method of completing the square, write this equation in the form y = a(x h) 2 + k. y x (b) Describe how the graph of y 2 x would be shifted to produce the graph of 2 y x x (c) Sketch the graph of point (vertex)? 2 y x x 8 18 by using its vertex form in (a). What are the coordinates of its turning 3

4 Example 2: Write each function in vertex form, and identify its vertex. Teacher Modeled Student Try it! y = x x 13 y = x 2 6x + 7 Step 1: Move the c term to the other side Step 2: Find the new c by using ( b 2 )2 and add to both sides Step 3: Write the perfect square trinomial as a binomial squared. Step 4: bring the old c term back over. y = y = Vertex/turning point: Vertex/turning point: Axis of symmetry: Axis of symmetry: Opening: Opening: Min/Max Value: 2 Describe how the graph of y x would be shifted to produce the graph above? Min/Max Value: Describe how the graph of produce the graph above? y 2 x would be shifted to 4

5 Regents Readiness.. 2) To Convert from y = ax 2 + bx + c Form to Vertex Form: y = a(x h) 2 + k. Method 1: Completing the Square Example 3: Write each function in vertex form, and identify its vertex. y = 2x 2-8x + 3 Step 1: Step 2: Step 3: Step 4: 1. move the c term to the other side. 2. Factor the a term out 3. Set up to complete the square. Find magic number that completes the square, and add to both sides. Don t forget to multiply by a!! 4. write as a binomial squared. 5. Move the "c" term back over. 5

6 Student Try it! f(x) = 5x x + 9 To Convert from y = ax 2 + bx + c Form to Vertex Form: y = a(x h) 2 + k. Method 2: Using the formula b x 2a Use the formula b x to find the turning points for each of the following quadratic functions. 2a y = 2x 2-8x + 3 step 1: x b 2a 1 2 g x x 5x 20 4 step 2: y = 2( ) 2 8( ) + 3 = Vertex: (, ) y = a(x h) 2 + k 6

7 SUMMARY: Exit Ticket

8 FLUENCY DAY 1 - COMPLETING THE SQUARE AND SHIFTING PARABOLAS COMMON CORE ALGEBRA II HOMEWORK 1. Which of the following equations would result from shifting y 2 x five units right and four units up? (1) y x (3) y x (2) y x (4) y x Which of the following represents the turning point of the parabola whose equation is 2 (1) 3, 7 (3) 7, 3 (2) 3, 7 (4) 3, 7 3. Which of the following quadratic functions would have a turning point at 6, 2? (1) y x (3) y x (2) y 3 x (4) y x y x 3 7? 4. Which of the following is turning point of (1) 12, 4 (3) 6,104 (2) 6, 40 (4) 4,12 2 y x x 12 4? 5. In vertex form, the parabola 2 y x x 10 8 would be written as (1) y x (3) y x (2) y x (4) y x 2 6. The turning point of the parabola y x x 5 2 is (1) 2.5,12.75 (3) 2.5, 8.25 (2) 5, 10.5 (4) 2.5,

9 7. Write each of the following quadratic functions in its vertex form by completing the square. Then, identify its turning point. (a) 2 y x x (b) y x x b 8. Use the formula x to find the turning points of each of the following quadratic functions. Then, place 2a the function in vertex form to verify the turning points. (a) y x x (b) y x x Consider the quadratic function whose equation is (a) Determine the y-intercept of this function algebraically. 2 y x x (b) Write the function in its vertex form. State the coordinates of its turning point. (c) Algebraically find the zeroes of the function using the zero product law. (d) Sketch a graph of the parabola, showing all relevant features found in parts (a) through (c). 9

10 Day 1 : Answer 10

11 Page 11

12 Day 2 - MORE PRACTICE WITH SHIFTING PARABOLAS SWBAT: Write a quadratic from standard form to vertex form. Warm Up Page 12

13 Page 13

14 Graphing Parabolas in Vertex Form Identify the vertex and axis of symmetry of each. Then sketch the graph. Page 14

15 Page 15

16 Answers to Day 2 Page 16

17 Page 17

18 Page 18

19 Day 3 - The Definition of the Parabola SWBAT graph a parabola with a directrix and focus. Page 19

20 1 2 Exercise #1: The parabola y x 1 is shown graphed below with selected points shown. For this parabola, 4 its focus is the point 0, 2 and its directrix is the x-axis. y How far is the turning point 0,1 from both the focus and directrix? How far is the point 2, 2 from both? Focus x Directrix Exercise #2: Refer to the graph. a) How far is the vertex from the directrix? b) What are the coordinates of the focus? Exercise #3: Refer to the graph. a) How far is the vertex from the directrix? b) What are the coordinates of the focus? Page 20

21 Exercise #4: Refer to graph: a) What are the coordinates of the vertex? b) What are the coordinates of the focus? c) What is the equation of the directrix? d) Find p, the distance between the vertex and the focus or the directrix? e) Let s write the equation of the parabola in conic vertex form: Page 21

22 Conic Vertex Form of a Parabola Page 22

23 Example 1: Graph the parabola with the equation y = 1 12 (x 1)2 + 2 Vertex: P: Focus: Directrix: The parabola opens. Page 23

24 Example 2: Graph the parabola with the equation y = 1 8 (x + 3)2-4 Vertex: P: Focus: Directrix: The parabola opens. Example 3: Graph the parabola with the equation x = 1 4 (y - 2)2 +4 Vertex: P: Focus: Directrix: The parabola opens. Example 4: Graph the parabola with the equation x = 1 12 y2-3 Vertex: P: Focus: Directrix: The parabola opens. Page 24

25 Graph each of the following parabolas. y = 1 8 (x 2)2 1 x = 1 12 y2 + 2 y = 1 4 (x + 3)2 x = 1 10 (y 1)2 + 3 Page 25

26 Day 3 Homework Page 26

27 Page 27

28 Answer Key: HW #3 Page 28

29 16. Page 29

30 Day 4: Writing Equations of Parabolas SWBAT write equations of parabolas in conic vertex form. Do Now: a) Given a parabola with a directrix at x = 1 and focus at (5,3), find the coordinates of the vertex. b) What is find the value of p? c) Can you write an equation of the parabola? Steps for Writing the Equation of a Parabola Page 30

31 Example 1: Write the equation of the parabola shown: P: Vertex: Equation: Example 2: Write the equation of a parabola with focus at (1,6) and a directix at y = 2. P: Vertex: Equation: Example 3: Write the equation of the parabola shown: P: Vertex: Equation: Page 31

32 Example 4: Write the equation of a parabola with focus at (-4,2) and a directix at x = 2. P: Vertex: Equation: Example 5: P: Vertex: Equation: Example 6: Write the equation of a parabola with vertex at (0,3) and the x axis as the directix. P: Focus: Equation: Page 32

33 Example 7: Write the equation of the parabola shown: P: Vertex: Equation: Page 33

34 Exit ticket: Fill in the following locus definition of a parabola with one of the words shown listed below. Words may be used more than once. point, line, equidistant, directrix, collection, focus A parabola is the of all points from a fixed and a fixed. The fixed is known as the parabola's. The fixed is known as the parabola's. Page 34

35 HW #4 # s odd and Pages all examples Page 35

36 Page 36

37 5. Page 37

38 Answer Key: Page 36 Page 35 Page 38

39 Answer Key: Page 37 Additional Questions: Set A Page 39

40 Additional Questions Set A Page 40

41 Answer Key: Set A Page 41

42 Additional Questions: Set B Page 42

43 Answer Key: Set B Page 43