1. Analyzing the Graphs of Functions
Objective: To use graphs to make statements about functions.

2. Finding Domain and Range of a Function
Use the graph to find:
The domain
The range
The values of f(-1), f(2)

3. Finding Domain and Range of a Function
Use the graph to find:
The domain
The range
The values of f(-1), f(2)
a) Domain = [-1, 5)
b) Range = [-3, 3]
c) f(-1) = 1; f(2) = -3

4. Vertical Line Test
A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

5. Vertical Line Test
A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.
We talked about this. A vertical line has the equation x = c. If this line intersects the graph in more than one place, that means for one value of x, there is more than one value for y.

6. Example 2
Use the vertical line test to decide whether the graphs represent y as a function of x.

7. Example 2
Use the vertical line test to decide whether the graphs represent y as a function of x.

8. Example 2
Use the vertical line test to decide whether the graphs represent y as a function of x.

9. Zeros of a Function
The zeros of a function f(x) are the x-values for which f(x)=0. This is what we did last chapter when we solved equations for 0. Graphically, we are finding the x-intercepts.

10. Example 3
Find the zeros of each function. a)

11. Example 3 Find the zeros of each function. a)
We need to find the zeros by
setting the equation equal to
zero and factoring.

12. Example 3 Find the zeros of each function. a)
We are now going to find the zeros with our calculator. 12

13. Example 3
Find the zeros of each function. b) 13

14. Example 3 Find the zeros of each function. b)
Again, we need to set the equation equal to zero and solve. A square root is equal to zero when the equation under the radical is equal to zero.

15. Example 3 Find the zeros of each function. b)
Again, we will use our calculator to find the zeros. 15

16. Example 3
Find the zeros of each function. c)

17. Example 3 Find the zeros of each function. c)
A fraction is equal to zero when
its numerator is equal to zero.

18. Example 3 Find the zeros of each function. c)
Again, let’s use the calculator 18

19. Relative Maximum/Minimum
A relative Maximum occurs at a peak, or a high point of a graph.
A relative Minimum occurs at a valley, or a low point of a graph.

20. Relative Maximum/Minimum
A relative Maximum occurs at a peak, or a high point of a graph.
A relative Minimum occurs at a valley, or a low point of a graph.
The term relative means that this is not the highest or lowest point on the entire graph, just at a certain place.

21. Relative Maximum/Minimum
A relative Maximum occurs at a peak, or a high point of a graph.
A relative Minimum occurs at a valley, or a low point of a graph.
We will be using our calculators to find these answers.

22. Increasing/Decreasing
A function is increasing when it is approaching a relative maximum.
A function is decreasing as it approaches a relative minimum.
Again, we will use our calculator to find these answers.

23. Increasing/Decreasing
Find where the function is increasing/decreasing.

24. Increasing/Decreasing
Find where the function is increasing/decreasing.
This function is increasing everywhere.
Increasing

25. Increasing/Decreasing
Find where the function is increasing/decreasing.

26. Increasing/Decreasing
Find where the function is increasing/decreasing.
Increasing
Decreasing

27. Increasing/Decreasing
Find where the function is increasing/decreasing.

28. Increasing/Decreasing
Find where the function is increasing/decreasing.
Increasing
Decreasing
Constant

29. Example 5
Use your calculator to find the relative minimum of the function and where the function is increasing or decreasing.

30. Example 5
Use your calculator to find the relative minimum of the function and where the function is increasing or decreasing.
So the relative minimum
is at the point (0.67, -3.33).
This function is decreasing
and increasing

31. Example 5
You try:
Find the relative max and min for the following function. Then, state where the function is increasing and decreasing.

32. Example 5
You try:
Find the relative max and min for the following function. Then, state where the function is increasing and decreasing.
Max (0, 4)
Min (2, -4)
Increasing
Decreasing