2. Tuesday Evaluate these two functions

3. Function Characteristics Even vs Odd Symmetry Concavity Extreme

4. Objectives I can prove a function is even, odd, or neither I can determine what type of symmetry a function has from a graph I can find extreme of a function (minimums/maximums) I can recognize concavity intervals based on inflection points

5. Symmetry Symmetry means that one point on the graph is exactly in the same position on the other side of the symmetric line. Graphs can symmetric with respect to: –x-axis –y-axis –A coordinate Point (Origin)

6. Section 1.2 : Figure 1.21, Symmetry Symmetric wrt y-axis

7. Graphical Tests for Symmetry 1. A graph is symmetric wrt the x-axis, if whenever (x, y) is on the graph, so is (x, -y) 2. A graph is symmetric wrt the y-axis, if whenever (x, y) is on the graph, so is (-x, y) 3. A graph is symmetric wrt the origin, if whenever (x, y) is on the graph, so is (-x, -y)

8. FUNCTIONSFUNCTIONS Symmetric about the y axis Symmetric about the origin

9. A function f is even if for each x in the domain of f, f (– x) = f (x). Even Functions x y f (x) = x 2 f (– x) = (– x) 2 = x 2 = f (x) f (x) = x 2 is an even function. Symmetric with respect to the y-axis.

10. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Even functions have y-axis Symmetry

11. A function f is odd if for each x in the domain of f, f (– x) = – f (x). Odd Functions x y f (x) = x 3 f (– x) = (– x) 3 = –x 3 = – f (x) f (x) = x 3 is an odd function. Symmetric with respect to the origin.

12. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Odd functions have origin Symmetry

13. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. x-axis Symmetry

14. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

15. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

16. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

17. Determine algebraically whether f(x) = –3x 2 + 4 is even, odd, or neither. Function Type Problems f(x) is an even function by definition. Is this function symmetrical?

18. Determine algebraically whether f(x) = 2x 3 - 4x is even, odd, or neither. Practice Problem Seven Is this function symmetrical? f(x) is an odd function by definition.

19. Practice Problem eight Determine algebraically whether f(x) = 2x 3 - 3x 2 - 4x + 4 is even, odd, or neither. Is this function symmetrical? f(x) is neither odd or even.

20. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Minimum and Maximum Values A function value f(a) is called a relative minimum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). x y A function value f(a) is called a relative maximum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). Relative minimum Relative maximum

21. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then at c the value of the function f is largest (at least in the area near there, hence “locally”). The value of c is called a local maximum of f. increasing here decreasing here f(-2) = 5 So 5 is called a local maximum of the function since for all x values close to –2, 5 is the maximum function value (y value).

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then at c the value of the function f is smallest (at least in the area near there, hence “locally”). The value of c is called a local minimum of f. increasing here decreasing here f(4) = -1 So -1 is called a local minimum of the function since for all x values close to 4, -1 is the minimum function value (y value).

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.22 Concavity A graph may be concave up or concave down See graphs below:

24. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Concavity Examples

25. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Concavity Examples

26. 25 Inflection Points An inflection point on a graph is where the graph changes concavity. –It changes from concave up to concave down –Or it changes from concave down to up

27. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Inflection Points

28. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Inflection Points

29. Homework WS 1-4 Quiz Next Class