1. Quadratic Functions, Quadratic Expressions, Quadratic Equations
Definition: A quadratic function is a function of the form
where a, b, c are real numbers and a  0.
The expression on the right-hand-side
is call a quadratic expression.

2. Quadratic Expressions: Factored Form
Examples: 1. 2.

3. Factoring quadratic expressions:
Given where a, b, c are integers.
Case 1: a = 1;
Since

4. we have

5. Examples:

6. Case 2: where a, b, c are
integers and a  1.
Since

7. we have

8. Examples:

9. Quadratic Equations:
A quadratic equation is an equation of the form:
Problem: Find the real numbers x, if any, that satisfy the equation.
The numbers that satisfy the equation are called solutions or roots.

10. Methods of Solution:
Method 1: Factor
then the solutions (roots) of the equation
are

11. Examples:

12. Method 2: Use the QUADRATIC FORMULA
The real number solutions (roots) of the quadratic equation
are:
provided
Method 2: Use the QUADRATIC FORMULA

13. The quadratic formula is often written as
The number is called the discriminant.

14. The Discriminant:
Given the quadratic equation
If:

15. (1) ; the roots are:
(2) the roots are:
(3) no real roots.

16. Examples:

19. Quadratic Functions:
The graph of is a parabola. The graph looks like
if a > if a < 0

20. Key features of the graph:
The maximum or minimum point on the graph is called the vertex. The x-coordinate of the vertex is:

21. The y-intercept; the y-coordinate of the point where the graph intersects the y-axis.
The y-intercept is:
The x-intercepts; the x-coordinates of the points, if any, where the graph intersects the x-axis. To find the x-intercepts, solve the quadratic equation

22. Examples:
Sketch the graph of
vertex:
y-intercept:
x-intercepts:

24. Sketch the graph of
Vertex:
y-intercept:
x-intercept(s):

26. Sketch the graph of
Vertex:
y-intercept:
x-intercept(s):