Algebra Second Six Weeks October 6 November 14, Monday Tuesday Wednesday Thursday Friday

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1 Algebra Second Six Weeks October 6 November 14, 014 Monday Tuesday Wednesday Thursday Friday October 6 B Day 7 A Day 8 B Day 9 A Day 10 B Day 3. Substitution and Elimination -from contexts, write equations, determine best method to use, define variables and solve. - variable 3.3 Substitution and Elimination -continue solving -3 variable (NEW) 3.4 Solving systems with matrices -Setting up matrices from systems of equations - solving systems with matrices A Day 15 B Day 16 A Day 17 B Day No School Professional Development Day Flex Day 3.5 Linear Programming -Writing Equations and inequalities -Graphing Restraints -Find critical points and calculate max and mins 0 A Day 1 B Day A Day 3 B Day 4 A Day Unit 3 Elaboration/Flex Day 4.1 Representations of Quadratic Functions -analyze & predict using quadratic data -make scatterplot -graph and look at tables -quadratic regression 7 B Day 8 A Day 9 B Day 30 A Day 31 B Day 4.1 Representations of Quadratic Functions -analyze & predict using quadratic data -make scatterplot -graph and look at tables -quadratic regression 4. Transforming Quadratics -up/down, left/right, horz/vert stretch/compression -domain/range in inequality, interval & set notaion Begin 4.3 PSAT UNIT 3 TEST 4.3 Three forms of a Quadratic -discuss factored, vertex, and standard form, their characteristics and uses -complete the square to move from standard to vertex Nov 3 A Day 4 B Day 5 A Day 6 B Day 7 A Day 4.4 Parabola Conic Sections -graph and identify key attributes (vertex, focus, directrix, axis of symmetry, direction of opening) -write equation given attributes Unit 4 Elaboration/Flex UNIT 4A TEST 10 B Day 11 A Day 1 B Day 13 A Day 14 B Day Reteach/Retest 4A 4.6 Solve by graphing -use calculator -solve linear & quadratic intersection(s) (NEW) -Look at zeros/roots/x-intercepts and solutions on a graph by hand and on calculator 1

2 Algebra 4. Transformations of Quadratic Name: Transformation Stations Worksheet Remember: Use the Vertex Form for a Quadratic Function translate Quadratic Functions from the Parent Function y = a( x h) + k y = x to Station 1: Station 4: Station : Station 5: Station 3: Station 6:

3 Algebra Name: 4. Transformations of Quadratic CONCLUSION: What happens when you change the value of.. a: h: k: y = a( x h) + k 3

4 Algebra 4. Transformations of Quadratic Name: For #1-4, Graph each of the following equations. State the domain, range and axis of symmetry for each. 1) y x 1 ) y x 1 Transformations: Transformations: D: R: AOS: D: R: AOS: 3 points (, ),(, ),(, ) 3 points (, ),(, ),(, ) 3) 3 1 y x 1 4) y x 3 Transformations: Transformations: D: R: AOS: D: R: AOS: 3 points (, ),(, ),(, ) 3 points (, ),(, ),(, ) 4 Page 1 of 4

5 Algebra 4. Transformations of Quadratic Name: 5. Which graph shows a function y = x + c when c < -1? 6. The graph of y = 11x + c is a parabola with a vertex at the origin. Which of the following is true about the value of c? a. c > 0 c. c = 0 b. c < 0 d. c = Shirley graphed a function of the form y = ax + c. She then translated the graph 8 units up, resulting in the function y x 5. 3 Which of the following best represents Shirley's original function? a. y x 13 3 b. y x 13 3 c. y x 3 3 d. y x If, how does the graph of compare to the graph of? a. The graph of is below the graph of. b. The graph of is above the graph of. c. The graph of is narrower the graph of. d. The graph of is wider the graph of. 5 Page of 4

6 Algebra 4. Transformations of Quadratic Name: 9. The graph of a function of the form is shown below. If the graph is translated only up or down to include the ordered pair (6, 7), which of the following equations best represents the resulting graph? a. c. b. d. 10. The grid below shows parabolas A and B of the form. How are the parabolas A and B related? a. Parabola A is narrower than parabola B. b. Parabola A is wider than parabola B. c. All the points on parabola A are 7 units below the corresponding points on parabola B. d. All the points on parabola A are 7 units above the corresponding points on parabola B. 6 Page 3 of 4

7 Algebra 4. Transformations of Quadratic Name: 11. Given the functions f(x) and g(x) as described in the following tables: f(x) g(x) X Y X Y Describe the transformation from f(x) to g(x) in f(x) form. 1. Given the functions f(x) and g(x) as described in the following tables: f(x) g(x) X Y X Y Describe the transformation from f(x) to g(x) in f(x) form. 13. Describe two different transformations that would make the parabola pass through the point (-, -3). 14. Given f ( x) x 6 write the equation of g(x) using the transformation g(x) = f(x + 3) 4. 7 Page 4 of 4

8 Algebra 4.3 Three Forms of a Quadratic Name: 1. A hawk is diving down to catch a rabbit and carry it back to his nest. The path is modeled by a quadratic. Three people have attempted to model the path and given you the following equations: Enter the functions into your calculator and graph: S(x) = x x 8 F(x) = x 4x V(x) = x 1 9 a) What do you notice about the three graphs? b) Verify algebraically that S(x) and F(x) are equal by solving the following equation: S(x) = F(x) x x 8 = (x 4)(x + ) c) Verify algebraically that S(x) and V(x) are equal. d) Find the following critical attributes: Vertex: Roots: y-intercept: 8

9 Each of the 3 forms of quadratic equations can be identified by the critical attributes above. Which equation tells you which attribute? Match them below. Standard Form S(x) Factored Form F(x) Vertex Form V(x) Vertex Roots y-intercept. Now an archer shoots his bow at a barbarian on Clash of the Clans, making a parabolic path. Graph the following three equations: F(x) = (x + 6)(x ) V(x) = (x + ) 3 S(x) = x + 8x 4 a) What value do you notice is the same in all three equations? b) List the critical attributes of the quadratic function and match each attribute with one of the equations. c) Verify algebraically that S(x) and F(x) are equal. d) Verify algebraically that S(x) and V(x) are equal. 9

10 Algebra 4.3 Three Forms of a Quadratic Name: The following is a picture of a Quadratic. Label the: Roots: (r 1, 0) and (r, 0) Vertex (max/min): (h, k) y-intercept: (0, c) Write the three forms of this Quadratic. Use a for your a value. These equations will be the general forms for Standard, Vertex, and Factored forms of a Quadratic. Standard form: Vertex form: Factored form: 10

11 Algebra Elaborate 4.3 Three Forms of a Quadratic Name: Graph the given data and then write the equation for the quadratic in the three forms we have learned. (F is factored form, V is vertex form, and S is standard form) 1) Vertex: (3, -4) Roots: (1, 0) and (5, 0) Y- intercept: (0, 5) a = 1 F(x) = V(x) = S(x) = ) A baseball is thrown from 5 feet above the ground. The ball reaches a maximum height of 9 feet after seconds. The ball lands on the ground 5 seconds after it was thrown. F(x) = V(x) = S(x) = 3) Given the following two graphs, write the three equations that represent the graphs. a) F(x) = V(x) = S(x) = 11

12 b) F(x) = V(x) = S(x) = 5) Given the following tables, what are the roots of the equation? x y ) Given the following tables, what are the roots of the equation? If there aren t any zeros in the y column, we can probably get a really good guess as to where the zeros are. x y

13 Algebra 4.3 Three Forms of a Quadratic Name: 1. The following equations are three forms of the same quadratic equation. S(x)= x x 15 F(x) = (x + 3)(x 5) V(x) = (x 1) 16 Find the following and identify the form of the equation that gives that information: a) y-intercept b) vertex c) roots. Rewrite the following equations in standard form: a) y = 3(x 1)(x + ) b) y = -1(x + 3) 4 c) y = (x 4) Tommy shot a bottle rocket off the top of his garage, which is 6 feet from the ground. The bottle rocket reaches a maximum height of 98 feet after six seconds and hits the ground after 13 seconds. a) Sketch a graph of the height as a function of time. Label the axes, vertex, y- intercept, and the positive x-intercept. b) Write the equation for this problem situation in the three forms. Be sure to find the a value within the equation. Factored Form: F(x) = Vertex Form: V(x) = Standard Form: S(x) = 13

14 4. Tiger Woods tees off from the top of a hill. A diagram of the situation is shown below on a coordinate grid. (, ) x = 160 (340, 0) a) Given a = -.005, write the equation in factored form. F(x) = b) Find the maximum height of the ball and label the vertex on the graph. c) Write the equation in the vertex and standard form. V(x) = S(x) = d) If the origin is considered the base of the hill, how high is the hill? 5. Given the following two graphs, write the three equations that represent the graphs (14, 11) (3, 0) (5, 0) (0, 0) (4, 0) (,-1) Factored Form: F(x) = Factored Form: F(x) = Vertex Form: V(x) = Vertex Form: V(x) = Standard Form: S(x) = Standard Form: S(x) = 14

15 Algebra 8.3 Explain: Parabolas PPT Parabola set of all points equidistant from a fixed line ( ) and a fixed point ( ). midpoint of segment from focus to directrix. line through focus and vertex Vertical Parabola Horizontal Parabola Form: Form: p: distance from vertex to and from vertex to. p p p p Write equation in standard form by completing the square. Ex1: y 4x 8x 8 0 Decide whether parabola has vertical or horizontal Axis of Sym. and which way the graph opens. Ex: -6x = 3y Ex3: (y 4) = 3x + 1 Ex4: y = (x + 1) Given the following information, write the equation of the parabola. 5. Vertex (-, 5); p = -1/; Vertical Axis of Symmetry 6. Vertex (1, -3); p = 1/8; Vertical Axis of Symmetry 7. Vertex (6, -1); p = -1/1; Horizontal Axis of Symmetry 8. Vertex (-5, -7); p = 1; Horizontal Axis of Symmetry Page 1 of 4 15

16 Algebra 8.3 Explain: Parabolas PPT Vertex: Focus: Directrix: Axis: Equation: Vertex: Focus: Directrix: Axis: Equation: Vertex: Focus: Directrix: Axis: Equation: Vertex: Focus: Directrix: Axis: Equation: Page of 4 16

17 Algebra 8.3 Explain: Parabolas PPT Find the vertex, focus, and directrix and sketch the graph y1 x x3 y 5 Vertex: Focus: Vertex: Focus: Directrix: Directrix: Find the equation, given the following information. 15. Vertex (-3, 6) Focus (5, 6) 16. Vertex (, 1) Directrix: x = Directrix: y = 5 Focus ( 3, 1) 18. Where are parabolas used? Page 3 of 4 17

18 Algebra 8.3 Explain: Parabolas PPT Axis of Symmetry: x = LR: Focus (, ) p Vertex: (, ) Directrix: y = ( x h) 4 p( y k) ( y k) 4 p( x h) Vertex (h, k) (h, k) Axis of symmetry x = h y = k Focus h, k p h p, k Directrix y k p x h p Direction of opening up/down left/right Length of Latus Rectum 1 (LR) a or 4p 1 or 4p a P 1 a 4 p 1 a 4 p Page 4 of 4 18

19 Algebra 8.3 Evaluate: Parabolas Name Write the equation of the parabola in standard form: 1. x x y y x y Decide whether the parabola has a vertical or horizontal axis: 3. 8x = y 4. 3x = 4y Find p, the focus, and the directrix of the parabola: y x 8( x1) y 6 6. Match the equation with the graph: 7. y = -x 8. x = y 9. x = -y A) B) C) f(x) f(x) x 1 x f(x) x 19

20 Algebra 8.3 Evaluate: Parabolas Name Graph the following equations: 10. ( y ) = 16( x 1) 11. ( x1) = -1( y 3) f(x) x f(x) x Write the equation of the parabola in standard form. 1. Vertex: (3, -) Focus: (5, -) 13. Vertex: (, ) Directrix: y = Steve Jobs has asked you to do some consulting on a secret project for Apple. The next ipod, the ipod wireless needs to have a parabola inside of it to communicate with the Apple satellite system. Mr. Jobs needs you to write the equation of a parabola with vertex at (5,1) and directrix x = 6. 0

21 Transforming Quadratics Using a, h, and k to transform Quadratics The quadratic parent function is. Its Vertex is. The quadratic function in the vertex form is the effect of each variable. (a, h, k). Describe (-): h: 0 < a < 1: k: a > 1: A. A quadratic reflected over the x axis B. Given the following function, give an Translated 4 units up and 6 units to example of a number that would make f ( x) 1 x 3 4 the left from the parent function. the graph wider. C. Use transformations to write the equation of the following graph. D. Given the functions f(x) and g(x) as described in the following tables: f(x) X Y g(x) X Y Describe the transformation from f(x) to g(x) in f(x) form. 1

22 Use patterns to write equations Representations of Quadratics x y x y How can we tell? It is Definitely Linear! It is Definitely Quadratic! A. List out the steps to writing an B. The following table shows the growth equation using the calculator of bacteria in a lab. Write the quadratic equation that best models this scenario C. Use stat regression to write the equation to D. Given the pattern, write the equation the following graph. 17, 8, 43, 6, 85

23 Graphing Quadratics A. Nicole shot her rocket into the air from the floor. It landed next to Kelly s desk B. Mrs. Ballero s pet lemur made the following jump. Give the reasonable range for the 1 seconds later. Write the equation. a=-16 function. Modeled by 1 3 y x 4x C. Write the equation to the axis of symmetry D. Reflect the following graph across the x axis. for the equation f x x x ( ) 15 3

24 Critical Attributes Use the your knowledge of quadratics to find the critical attributes V( x) a( x h) k s( x) ax bx c f ( x) a( x r1)( x r) What do I know from this? What do I know from this? What do I know from this? A. Write the equation to the following scenario Hit over a 10 foot fence after 50 secs. He hit from a height of 6 feet. B. The following equation is in the vertex form. Write it in standard form. x V ( x) 4 3 C. What is the x value to the max of the graph that has the roots (-,0) and (6, 0)? D. Write the 3 forms of the quadratic from the table. X Y

25 Factoring Quadratic y ax bx c Find the multiples of ac. Find the that sum to be b. Put in the parenthesis as such (x+ )(x+ ). You Factored! A. Factor x 7x 1 B. Factor f x ( ) x 100 C. Factor to put in standard form and give the D. Factor to put in standard form and give roots. y x x 0 the roots. y x x 5 6 5

26 3 Names of the Quadratic Use the 3 forms to represent quadratics V( x) a( x h) k f ( x) a( x r1)( x r) s( x) ax bx c What do I know from this? What do I know from this? What do I know from this? A. Circle the part of each form that tells if the parabola is concave up or down. V( x) a( x h) k B. What are the roots to the following function 19 f ( x) x 4 x s( x) ax bx c f ( x) a( x r )( x r ) 1 C.Use the table below to find the information. X Y Vertex: Roots: D. Write the 3 forms of the quadratic for the graph below. 6