Properties of Functions Dr. Fowler AFM Unit 1-3 Properties of Functions
Even and Odd Functions (graphically) If the graph of a function is symmetric with respect to the y-axis, then it’s even. If the graph of a function is symmetric with respect to the origin, then it’s odd. The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then it’s Odd. If you get different y’s, then it’s Neither. Copyright © by Houghton Mifflin Company, 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.
VIDEO - Odd, Even, or Neither Function? https://www.youtube.com/watch?v=1LsJaR72UFM
Even and Odd Functions (algebraically) A function is even if f(-x) = f(x) If you plug in -x and get the original function, then it’s even. The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then it’s Odd. If you get different y’s, then it’s Neither. A function is odd if f(-x) = -f(x) If you plug in -x and get the opposite function, then it’s odd. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
EVEN Example 1 Even, Odd or Neither? Graphically Algebraically Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
ODD Example 2 They are opposite, so… Even, Odd or Neither? Graphically What happens if we plug in 2? Graphically Algebraically ODD They are opposite, so… Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Neither Example 4 Even, Odd or Neither? Graphically Algebraically Copyright © by Houghton Mifflin Company, 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
The local maximum value is 2. Y values Local maximum when x = 1. Before a turn back down. The local maximum value is 2. Y values
The local minima values are 1 and 0. (y values) Local minimum when x = –1 and x = 3. X values before a turn up. The local minima values are 1 and 0. (y values)
(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
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 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
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
Excellent Job !!! Well Done