Properties of Functions

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Properties of Functions Section 2.3 Properties of Functions Copyright © 2013 Pearson Education, Inc. All rights reserved

Determine even and odd functions from a graph. Objectives: Determine even and odd functions from a graph. Identify even and odd functions from the equation. Use a graph to determine where a function is increasing, decreasing or constant. Use a graph to locate local maxima and local minima. Use a graph to locate the absolute maximum and the absolute minimum. Find the average rate of change of a function. Copyright © 2013 Pearson Education, Inc. All rights reserved

For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Copyright © 2013 Pearson Education, Inc. All rights reserved

So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd. Even function because it is symmetric with respect to the y-axis Neither even nor odd because no symmetry with respect to the y-axis or the origin. Odd function because it is symmetric with respect to the origin. Copyright © 2013 Pearson Education, Inc. All rights reserved

Odd function symmetric with respect to the origin Even function symmetric with respect to the y-axis Since the resulting function does not equal f(x) nor –f(x) this function is neither even nor odd and is not symmetric with respect to the y-axis or the origin. Copyright © 2013 Pearson Education, Inc. All rights reserved

Where is the function increasing? Copyright © 2013 Pearson Education, Inc. All rights reserved

Where is the function decreasing? Copyright © 2013 Pearson Education, Inc. All rights reserved

Where is the function constant? Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

There is a local maximum when x = 1. The local maximum value is 2 Copyright © 2013 Pearson Education, Inc. All rights reserved

There is a local minimum when x = –1 and x = 3. The local minima values are 1 and 0. Copyright © 2013 Pearson Education, Inc. All rights reserved

(e) List the intervals on which f is increasing. (f) List the intervals on which f is decreasing. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the absolute maximum and the absolute minimum, if they exist. The absolute maximum of 6 occurs when x = 3. The absolute minimum of 1 occurs when x = 0. Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the absolute maximum and the absolute minimum, if they exist. The absolute maximum of 3 occurs when x = 5. There is no absolute minimum because of the “hole” at x = 3. Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the absolute maximum and the absolute minimum, if they exist. The absolute maximum of 4 occurs when x = 5. The absolute minimum of 1 occurs on the interval [1,2]. Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the absolute maximum and the absolute minimum, if they exist. There is no absolute maximum. The absolute minimum of 0 occurs when x = 0. Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the absolute maximum and the absolute minimum, if they exist. There is no absolute maximum. There is no absolute minimum. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

a) From 1 to 3 Copyright © 2013 Pearson Education, Inc. All rights reserved

b) From 1 to 5 Copyright © 2013 Pearson Education, Inc. All rights reserved

c) From 1 to 7 Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

-4 3 2 -25 Copyright © 2013 Pearson Education, Inc. All rights reserved

Homework: Section 2.3 (page 89): #11-51 odd, 61, 65 Copyright © 2013 Pearson Education, Inc. All rights reserved 33

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