Algebra 2A – Unit 2

1. Algebra 2A – Unit 2 Quadratic Functions (Graphing)

2. 2-1: Graphing Quadratic FunctionsLearning Targets: I can graph quadratics using five points. I can find and interpret maximum and minimum values.

3. Graph of a quadratic function: Plug x into the function to find y. This is where the graph makes its turn.

4. Using the TI to Graph Know these function keys on your calculator: y = set window table max and min

5. Example 1:Graph f(x)= 3x2 - 6x+ 7 Axis of symmetry: Direction of opening: Up / down? “a” value? Maximum / Minimum? Value: _____________ Minimum of 4 @ x=1 Now plot the points: -1 2 3 0 1 (1, 4) 16 7 4 7 16

6. Example 2:Graph f(x)= - x 2 + 6x- 4 Now plot the points: Axis of symmetry: Direction of opening: Up / down? “a” value? Maximum / Minimum? Value: _____________ Maximum of 5 @ x=3 1 4 5 2 3 (3, 5) 1 4 5 4 1

7. Lesson 2.1: Closing Can you graph quadratics using five points and/or your calculator? H.W. Practice 2.1 Can you find max. and min. values?

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11. Algebra 2A – Unit 2Lesson 2.2 Analyzing graphs of Quadratic Functions

12. Lesson 2.2Learning Targets: I can graph quadratic functions in vertex form. I can write quadratic functions in standard and/or vertex form.

13. Vertex Form Standard Form

14. Exploration Activity

15. Example 1:Graph f(x)= (x – 3)2 – 2 Now plot the points:

16. Example 2:Graph f(x)= ½ (x + 5)2 + 3

17. Your Turn 1:Graph f(x)= 4(x + 3)2 – 2

18. Your Turn 2:Graph f(x)= -2 (x + 1)2 + 4

19. Example 3: • Plug in values for h, k and x, y to find a. • Now look at this: Rewriting the equation in standard form. -20 = a(0 – 4)2 - 36 • Next, solve for a. y = 1(x – 4)2 - 36 -20 = 16a - 36 16a = 16 y = 1(x– 4)(x – 4) - 36 a = 1 y = 1(x2 -8x +16)- 36 • Finally, rewrite the equation using a, h and k. y = x2 -8x - 20 y = 1(x – 4)2 - 36 Same parabola in different forms.

20. Example 4: • Plug in values for h, k and x, y to find a. 2 = a(0+3)2 - 1 • Next, solve for a. 2 = 9a - 1 3 = 9a • Finally, rewrite the equation using a, h and k.

21. Your Turn 3: • Plug in values for h, k and x, y to find a. 0 = a(2 - 3)2 +- 1 • Next, solve for a. 0= a - 1 1 = a • Finally, rewrite the equation using a, h and k. y = 1(x - 3)2 - 1

22. Use your Graphing Calculator to solve the following problems: Word Problem 1:An object is propelled upward from the top of a 500 foot building. The path that the object takes as it falls to the ground can be modeled by h = -16t2 +100t + 500 where t is the time (in seconds) and h is the corresponding height of the object. The velocity of the object is v = -32t +100 where t is seconds and v is velocity of the object y = How high does the object go? ___________________ When is the object 550 ft high? __________________ With what velocity does the object hit the ground? __________________

23. Use your Graphing Calculator to solve the following problems: Word problem 2: An astronaut standing on the surface of the moon throws a rock into the air with an initial velocity of 27 feet per second. The astronaut’s hand is 6 feet above the surface of the moon. The height of the rock is given by h = -2.7t2+ 27t + 6 y = How many seconds is the rock in the air? How high did the rock go?

24. Lesson 2.2: Assignment • Check for Understanding: Closure • Assignment: Practice 2.2

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28. Algebra 2A – Unit 2Lesson 2.3 Solving Quadratic Equations by graphing

29. Lesson 2.3Learning Targets: I can solve quadratic equations by graphing. (exact roots) I can estimate solutions of quadratic equations by graphing. (approximate roots)

30. Vocabulary x- intercepts of the graph solutions of the quadratic equation Cases: two real roots one real root no real roots , Ø

31. Example: Not a point but the 2 places where the graph crosses the x-axis. This is called a double root. -3 is the answer twice.

32. Old School or Vintage style: Part 1 : Exact rootsExample 1: Solve x2 + x – 6 = 0 by graphing. 2 -1 0 -2 -3 1 0 0 -6 -4 -4 -6 Exact Roots of the equation (or zeros of the function):________ & _______ -3 2

33. Part 1 : Exact rootsYour Turn 1: Old School or Vintage style: Solve x2 - 4x– 5 = 0 by graphing. 2 5 1 3 0 -1 4 0 0 -8 -9 -8 -5 -5 Exact Roots of the equation (or zeros of the function):________ & _______ -1 5

34. Use your Graphing Calculator to solve the following problems: Part 2 : Approximate rootsExample 2: Solve x2 - 2x– 2 = 0 by graphing. Look under the table in the calculator. 1 0 2 -1 3 1 -2 1 -3 -2 Approximate roots: _____________________________

35. Use your Graphing Calculator to solve the following problems: Part 2 : Approximate rootsYour Turn 2: Solve x2 - 4x+ 4 = 0 by graphing. 2 1 3 0 4 4 1 4 0 1 Approximate roots: _____________________________ Double Root of x=2

36. Their sum is 4, and their product is -12. Use your Graphing Calculator to graph and find the roots:

37. Lesson 2.3: Assignment • Check for Understanding: Closure • Assignment: Practice 2.3

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