1. 1. Use the discriminant to determine the number and type of roots of: a. 2x 2 - 6x + 16 = 0b. x 2 – 7x + 8 = 0 2. Solve using the quadratic formula: -3x 2 – 8x + 5 = 0 3. Which method would you use to solve this and why? 8x 2 - 32 = 0 4. Solve using the square root method (2x-1) 2 =6

2. February 17 th, 2015

4.  Sometimes we will be given data in a table or a list of points and asked to write the equation

5.  STAT - #1 Edit  Enter X values in L1  Enter Y values or f(x) in L2  STAT CALC  CALC - #5 QuadReg  Go down to Calculate  Put in the “a” “b” “c” values to form a quadratic

6. a = ______ b = ______ c = ________ Equation: (–2, 1), (–1, 0), (0, 1), (1, 4), (2, 9)

7. a = ______ b = ______ c = ________ Equation:

8.  The following data forms a parabola, what are the roots? Find the equation. xy -48 -30 -2-6 0-12 40 58

9.  Find the zeros, write out the factors, and multiply out!

10. What if the parabola opens down?

11.  If the equation does not have whole number zeros, you can always make a table of points from the graph and find the Quadratic Equation of best fit using “QuadReg” on the calculator

12.  Last year you studied systems of linear equations.  You learned three different methods to solve them.  Elimination, Substitution and Graphing

13.  To solve is to find the intersections of the graph.  Put each in slope intercept form and graph  This is what we will use to solve a Quadratic/Linear System

14. 1. Type equation 1 = y 1, equation 2 = y 2 2. Push 2 nd  TRACE  5, move cursor to the left intersection push ENTER 3 times. 3. Push 2 nd  TRACE  5, move cursor to the right intersection push ENTER 3 times.

18.  We do this the same way we do linear equations  To solve is to find the intersections of the graph.

20. How can we CHECK our answers?

24.  We can create quadratic functions to model real world situations all around us.  We can use these models to find out more information, such as:  Minimum/maximum height  Time it takes to reach the ground  Initial height  How long it takes to reach a height

27. For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. a) What is the maximum height of the ball? How long does it take to reach the maximum height? How do we approach this problem…

28.  Are we looking for x or for y?  Graph the function. Adjust the window as needed. (this takes some practice!)  Find the vertex.

29.  The maximum or minimum HEIGHT is represented by the Y VALUE of the vertex.  How long it takes to reach the max/min height is represented by the X VALUE of the vertex.

30. For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. b) When will the shot reach the height of the basket? (10 feet) How do we approach this problem…

31.  Set the equation equal to the height you want to be at  Let y 2 = given height  Let y 1 = the original equation  Find the intersection of y 1 and y 2

32.  The X VALUE always represents TIME How long it takes….  So when you find the intersection, it should have X = time, and Y = height

33. For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. c) When will the ball hit the floor if it missed the basket entirely? How do we approach this problem…

34.  This is asking us when does the height = 0  Let y 2 = 0.  Find the intersection of y 1 and y 2

35.  When asking when something HITS the GROUND you should think ZERO!  GROUND = ZERO  Find the second zero (not the first!) think left to right…goes up then down

36. For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. d) What is the height of the ball when it leaves the player’s hands? How do we approach this problem..

37.  Here we want to find the INITIAL HEIGHT….where did the ball start? ON the ground? In someone's hands?  The INITIAL HEIGHT is the Y- INTERCEPT!

38. For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. d) What is the height of the ball after 2 seconds? How do we approach this problem..

39.  I take the x-value (time) and plug it in to find the y-value (height)  h(2) = -16(2) 2 + 40(2) + 6 = ____ feet

40.  The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. a) How long does it take the diver to reach her maximum height after diving off the platform? b) When will the diver reach a height of 2 feet? c) What is her maximum height? d) When will the diver hit the water? e) How high is the diver after 1.5 seconds? f) How high is the diving board?

41.  The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t 2 + 80t. a) When will the rocket hit the ground? b) What is the highest point that the rocket reaches? c) When does it reach the highest point? c) At what time(s) is the rocket at a height or 25 m? d) What was the initial height of the rocket?

43.  Reasoning What are the solutions of the system y = 2 x 2 – 11 and y = x 2 + 2 x – 8? Explain how you solved the system.

44.  Begin working on the Practice Test for Quadratics!  Due on Thursday (TEST DAY)