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

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

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

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.

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.

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

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

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

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.

Example 3 Find the zeros of each function. a)

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

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

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

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.

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

Example 3 Find the zeros of each function. c)

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

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

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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