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Analyzing the Graphs of Functions

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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


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