1. How can I analyze graphs of FUNctions?

2. The Graph of a FUNction
Graphing an f(x) FUNction is no different than graphing equations in two variables.

4. The Vertical Line Test
If every vertical line meets the graph in at most one point, then the graph is a FUNction.
Are the following FUNctions?

5. Yes!

6. No!

7. Please find the domain of

8. Increasing and Decreasing FUNctions see GSP page 1 and 2
You already have the intuition to understand the following:
A FUNction f is increasing on an interval if, for any x1 and x2 in the interval such that x1 < x2, then f(x1) < f(x2)
A FUNction f is decreasing on an interval if, for any x1 and x2 in the interval such that x1 < x2, then f(x1) > f(x2)
A FUNction f is constant on an interval if, for any x1 and x2 in the interval, then f(x1) = f(x2)
see GSP page 3

9. Relative Minimum and Maximum Values
f(a) is a relative minimum of the function f if there exists an open interval I, containing a, such that f(a) ≤ f(x) for all x in I.
f(a) is a relative maximum of the function f if there exists an open interval I, containing a, such that f(a) ≥ f(x) for all x in I.
see GSP page 4

10. 5. The profit for a new company can be modeled by
P = 0.225x3 – 17.21x x
where P is in thousands of dollars and x is the number of units sold in thousands.
What would be the maximum profit for this company?
$1,822,680

11. Graphing Step FUNctions
Define the greatest integer function, denoted by [|x|], to be [|x|] = the greatest integer less than or equal to x (see page 93 for more attractive notation)
This is commonly referred to as a step function. Also, it is the type of function telephone companies use to bill us for long distance calls.
Graph this function with a graphing utility in the dot mode, not connected mode

12. Graphing Piecewise FUNctions
We graph piecewise FUNctions by hand, sketching each equation and erasing the parts that we do not want.
See GSP p 5

13. Even and Odd FUNctions
See GSP Odd and Even Presentation