1. Describing Graphs and Tables of Functions
Mrs. Viney's Website: Module Five Helps

2. If we examine a typical graph the function
y = f(x), we can observe that for an interval throughout which the function is defined, that the function might be increasing, decreasing or neither.

3. We say that a function is increasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) < f(x2).
Further, we say that f(x) is increasing at x = c provided that f(x) is increasing in some open interval on the x-axis that contains c.

9. We say that a function is decreasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) > f(x2).
Further, we say that f(x) is decreasing at
x = c provided that f(x) is decreasing in some open interval on the x-axis that contains c.

18. Extreme Points
A relative extreme point ( relative maximum point or relative minimum point) of a function is a point at which its graph changes from increasing to decreasing or vice versa.

19. A relative maximum point is a point at which the graph changes from increasing to decreasing.

20. A relative minimum point is a point at which the graph changes from decreasing to increasing.

21. The maximum value of a function is the largest value that the function assumes on its domain.
The minimum value of a function is the smallest value that the function assumes on its domain.

22. Note: Functions might or might not have maximum and/or minimum values.

24. Changing slope
Consider the next two graphs. Note that the graphs of both are increasing, but there is a difference in how they are increasing.
What is the difference?

25. Graph I

26. Graph II

27. We note that the slope of graph I is increasing while the slope of graph II is decreasing.
In application, we would say that the debt per capita depicted in graph I is rising at an increasing rate.
From graph II, we observe that the population is increasing at a declining rate.

28. Intercepts
We have previously discussed the idea of intercepts. Recall that
The x-intercept is a point at which a graph intersects the x-axis. (x,0)
The y-intercept is a point at which the graph intersects the y-axis. (0,y)

30. Note that a function can have at most one
y-intercept. Otherwise, its graph would violate the vertical line test for a function.
A function may have 0 or more x-intercepts.

31. We now have six categories for describing the graph of a function
Intervals in which the function is increasing or decreasing
Maximum/Minimum values
Domain
Range
x-intercepts, y-intercept
Continuous or Discrete

32. Describe This Function on the interval from (-4,3)

33. 1. Inc (-4, -1.5]
Dec [-1.5, 1.5] Inc [1.5, 3)
Max (-1.5, 13) Relative Maximum is 13 Min (1.5, -1) Relative Minimum is -1
Domain (-4, 3)
Range (-4, 16)
y-intercept = 6 (0,6) x-intercepts = -3, 1, 2 (-3,0) (1,0) (2,0)
Continuous

37. HW 5.2 #1-15
Mrs. Viney's Website: Module Five Helps