1. Chapter 5 Quadratic Functions & Inequalities

2. 5.1 – 5.2 Graphing Quadratic Functions The graph of any Quadratic Function is a Parabola To graph a quadratic Function always find the following: y-intercept (c - write as an ordered pair) equation of the axis of symmetry x = vertex- x and y values (use x value from AOS and plug in for y) roots (factor) These are the solutions to the quadratic function minimum or maximum domain and range If a is positive = opens up (minimum) If a is negative = opens down (maximum)

3. f(x) = x 2 + 2x - 3 Ex: 1 Graph by using the vertex, AOS and a table

4. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range f(x) = -x 2 + 7x – 14

5. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range f(x) = 4x 2 + 2x - 3

6. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range x 2 + 4x + 6 = 0

7. 2x 2 – 7x + 5 = 0 Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range

8. 5.7 Analyzing graphs of Quadratic Functions Most basic quadratic function is y = x2 Axis of Symmetry is x = 0 Vertex is (0, 0) A family of graphs is a group of graphs that displays one or more similar characteristics! y = x 2 is called a parent graph

9. Vertex Form y = a(x – h) 2 + k Vertex: (h, k) Axis of symmetry: x = h a is positive: opens up, a is negative: opens down Narrower than y = x 2 if |a| > 1, Wider than y = x 2 if |a| < 1 h moves graph left and right - h moves right + h moves left k moves graph up or down - k moves down + k moves up

10. Identify the vertex, AOS, and direction of opening. State whether it will be narrower or wider than the parent graph y = -6(x + 2) 2 – 1 y = (x - 3) 2 + 5 y = 6(x - 1) 2 – 4 y = - (x + 7) 2

11. Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = 4(x+3) 2 + 1

12. y = -(x - 5) 2 – 3

13. y = ¼ (x - 2) 2 + 4

14. 5.8 Graphing and Solving Quadratic Inequalities 1. Graph the quadratic equation as before (remember dotted or solid lines) 2. Test a point inside the parabola 3. If the point is a solution(true) then shade the area inside the parabola if it is not (false) then shade the outside of the parabola

15. Example 1: Graph y > x 2 – 10x + 25

16. Example 2: Graph y < x 2 - 16

17. Example 3: Graph y < -x 2 + 5x + 6

18. Example 4: Graph y > x 2 – 3x + 2

19. 5.4 Complex Numbers Let’s see… Can you find the square root of a number? A. F.E.D. C.B. G.

20. So What’s new? To find the square root of negative numbers you need to use imaginary numbers. i is the imaginary unit i 2 = -1 i = Square Root Property For any real number x, if x 2 = n, then x = ±

21. What about the square root of a negative number? E.D. C. B. A.

22. Let’s Practice With i Simplify -2i (7i) (2 – 2i) + (3 + 5i) i 45 i 31 A. B. C. D. E.

23. Solve 3x 2 + 48 = 0 4x 2 + 100 = 0 x 2 + 4= 0 A. B. C.

24. 5.4 Day #2 More with Complex Numbers Multiply (3 + 4i) (3 – 4i) (1 – 4i) (2 + i) (1 + 3i) (7 – 5i) (2 + 6i) (5 – 3i)

25. *Reminder: You can’t have i in the denominator Divide 3i5 + i 2 + 4i 2i -2i4 - i 3 + 5i 5i 2 + i 1 - i E. D. C. B. A.

26. 5.6 The Quadratic Formula and the Discriminant The Quadratic Formula: Use when you cannot factor to find the roots/solutions

27. The discriminant: the expression under the radical sign in the quadratic formula. *Determines what type and number of roots b 2 – 4ac DiscriminantType and Number of Roots b 2 – 4ac > 0 is a perfect square2 rational roots b 2 – 4ac > 0 is NOT a perfect square2 irrational roots b 2 – 4ac = 01 rational root b 2 – 4ac < 02 complex roots

28. Example 1: x 2 – 3x – 40 = 0 Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula

29. Example 2: 2x 2 – 9x + 15 = 0 Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula

30. Example 3: x 2 + 6x – 9 = 0 Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula

31. TOD: Solve using the method of your choice! (factor or Quadratic Formula) A. 7x 2 + 3 = 0B. 2x 2 – 5x + 7 = 3 C. 2x 2 - 5x – 3 = 0 D. -x 2 + 2x + 7 = 0