2. Tuesday: Welcome Back Today you will need to : 1. Find your new seat 2. Pick-up notes on file cabinet 3. Get out notebooks

3. Evaluate these two functions at (-x)

4. Function Characteristics Even vs Odd Types of Symmetry Concavity Extrema

5. 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 extrema of a function (minimums/maximums) I can recognize concavity intervals based on inflection points

6. 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 –Origin

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

8. Graphs of symmetry

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

11. 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.

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 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

13. 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.

14. 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

15. 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

16. Another Way to Remember Easy parent functions:

17. 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

18. 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

19. 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.

20. 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?

21. 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.

22. 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.

23. Extrema: (Minimums/Maximums) 2 Types Relative (Local) Extrema Absolute Extrema There can be many relative extrema on a given graph There can at most be one absolute maximum and at most one absolute minimum

24. 23 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

25. 24 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).

26. 25 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).

27. Graphs with extrema

29. 28 Concavity A graph may be concave up or concave down See graphs below:

30. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Concavity Examples

31. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 Concavity Examples

32. 31 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

33. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32 Inflection Points

34. 33 Inflection Points

37. Homework WS 1-4 Quiz Next Class