1. Quadratic Functions & Inequalities
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 solve 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) – y coordinate of the vertex
If a is negative = opens down (maximum) – y coordinate of the vertex

3. Ex: 1 Graph by using the vertex, AOS and a table
f(x) = x2 + 2x - 3

4. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range
f(x) = -x2 + 7x – 14

5. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range
f(x) = 4x2 + 2x - 3

6. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range
x2 + 4x + 6 = f(x)

7. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range
2x2 – 7x + 5 = f(x)

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 = x2 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 = x2 if |a| > 1, Wider than y = x2 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
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
Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points.
y = 4(x+3)2 + 1

12. Graph after identifying the vertex, AOS, and direction of opening
Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points.
y = -(x - 5)2 – 3

13. Graph after identifying the vertex, AOS, and direction of opening
Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points.
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 > x2 – 10x + 25

16. Example 2: Graph y < x2 - 16

17. Example 3: Graph y < -x2 + 5x + 6

18. Example 4: Graph y > x2 – 3x + 2

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

20. So What’s new? For any real number x, if x2 = n, then x = ±
To find the square root of negative numbers you need to use imaginary numbers.
i is the imaginary unit
i2 = -1
i =
Square Root Property
For any real number x, if x2 = n, then x = ±

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

22. Let’s Practice With i Simplify -2i (7i) (2 – 2i) + (3 + 5i) i45 i31 A.B. C. D. E.

23. Solve
3x = 0
4x = 0
x2 + 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
3i i
2 + 4i i
-2i i
3 + 5i i
2 + i
1 - i A. D. E. B. C.

26. 5.6 The Quadratic Formula and the Discriminant
The discriminant: the expression under the radical sign in the quadratic formula. *Determines what type and number of roots
Discriminant
Type and Number of Roots
b2 – 4ac > 0 is a perfect square
2 rational roots
b2 – 4ac > 0 is NOT a perfect square
2 irrational roots
b2 – 4ac = 0
1 rational root
b2 – 4ac < 0
2 complex roots

27. The Quadratic Formula:
Use when you cannot factor to find the roots/solutions

28. Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula
Example 1:
x2 – 3x – 40 = 0

29. Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula
Example 2:
2x2 – 8x + 11 = 0

30. Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula
Example 3:
x2 + 6x – 9 = 0

31. TOD: Solve using the method of your choice
TOD: Solve using the method of your choice! (factor or Quadratic Formula)
A. 7x2 + 3 = 0 B. 2x2 – 5x + 7 = 3
C. 2x2 - 5x – 3 = 0 D. -x2 + 2x + 7 = 0