1. Graphs and Functions (Review) MATH 207

2. Distance Formula Example: Find distance between (-1,4) and (-4,-2). Answer: 6.71

3. Midpoint Formula Example: Find the midpoint from P1(-5,5) to P2(-3,1). Answer: (-4,3)

4. The standard form of an equation of a circle with radius r and center (h, k) is: The Unit Circle equation is: x y (h, k) r (x, y) Equations in two variables – Example: Circle Equations

5. Definition of a Function

6. Theorem: Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

7. x y Not a function.

8. x y Function.

11. (a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain. (b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range. (c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x. Summary Important Facts About Functions

13. Properties of Functions: Even and Odd Functions A function f is even if for every number x in its domain the number -x is also in its domain and f(-x) = f(x) A function f is odd if for every number x in its domain the number -x is also in its domain and f(-x) = - f(x)

15. Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.

17. Where is the function increasing?

18. Where is the function decreasing?

19. Where is the function constant?

21. Local Maxima and Minima Local Max

22. Local Min

23. Average rate of change of a Function

24. From 0 to 1

25. Library of Functions (Famous Functions)

36. Piecewise-defined Functions: Example:

37. Application problem:

38. Graphing Functions:

53. . The inverse of f, denoted by f -1, is a function such that f -1 (f( x )) = x for every x in the domain of f and f(f -1 (x))=x for every x in the domain of f -1 : Inverse Functions

54. Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

55. y = x (2, 0) (0, 2)