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. y = -6(x + 2)2 – 1 y = (x - 3)2 + 5 y = 6(x - 1)2 – 4 y = - (x + 7)2
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. B. E. C.

26. 5.5 Completing the Square
To complete the square for any quadratic expression of the form 𝑥 2 + 𝑏𝑥 , follow the steps listed.
Step 1 Find one half of b, the coefficient of x.
Step 2 Square the result in Step 1.
Step 3 Add the result of Step 2 to 𝑥 2 + 𝑏𝑥 .
𝑥 2 + 𝑏𝑥 + ( 𝑏 2 ) 2 = (𝑥+ 𝑏 2 ) 2 or
𝑥 2 − 𝑏𝑥 + ( 𝑏 2 ) 2 = (𝑥− 𝑏 2 ) 2

27. 5.5 Completing the Square
Let’s try some:
Solve:

28. 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

29. 5.6 The Quadratic Formula and the Discriminant
Use when you cannot factor to find the roots/solutions

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

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

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

33. 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