Symmetry and Coordinate Graphs Section 3.1

1. Symmetry and Coordinate Graphs Section 3.1

2. Symmetry with Respect to the Origin Symmetric with the origin if and only if the following statement is true: F(-x)=-F(x)

3. Symmetric with Origin Example F(X) = x5 Yes F(x) = x/(1-x) No

4. Symmetry (a,b) X-Axis Plug in (a,-b) Y-Axis Plug in (-a,b) Y=X Plug in (b,a) Y=-X Plug in (-b,-a)

5. Example Determine whether the graph of xy=-2 is symmetric with respect to the x axis, yaxis, the line y=x, and the line y=-x, or none of these? First plug in (a,b) Ab=-2 Symmetric with both line y=x and line y=-x

6. Example Determine whether the graph of │y│ =│x│+1 is symmetric with respect to the x axis, yaxis, both or neither? Symmetric with both the x and y axis.

7. Even and Odd Functions Even  Symmetric with respect to Y axis F(-x)=F(x) Odd  Symmetric with respect to the origin F(-x)=-F(x) F(X) = x5 Odd F(x) = x/(1-x) Neither odd nor even

8. Even and Odd Functions Which lines are lines of symmetry for the graph of x2=1/y2 X and y axises, y=x, and y=-x Is the following function symmetric about the origin? F(X)=-7x5 + 8x Yes, does this mean it’s even or odd? Odd

9. Families of Graphs Section 3.2

10. Parent Graphs • Constant

11. Parent Graphs

12. Example • Graph f(x) = x2 and g(x) = -x2. Describe how the graphs of f(x) and g(x) and are related.

13. Changes to Parent Graph • Graph Parent Graph of f(x)=|x| • Graph f(x)=|x|+1 • Graph f(x) = |x|-1 • Graph f(x)=|x+1| • Graph f(x) = |x-1| • On same graph • Similarities/Differences?

14. Change to Parent Graph Reflections Y=-f(x) Outside the HVertical Axis Reflected over the x-axis Y=f(-x) Inside the HHorizontal Axis Reflected over the y-axis

15. Change to Parent Graph Translations +,- OUTSIDE of Function Outside the H Vertical Movement SHIFTS UP AND DOWN +,- INSIDE of Function Inside the H Horizontal Movement SHIFTS LEFT AND RIGHT

16. Change to Parent Graph Dilations X/÷ OUTSIDE of Function Outside the H Vertical Movement Expands/Compresses X/÷ INSIDE of Function Inside the H Horizontal Movement Expands/Compresses

17. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = x2 + 1 • This function is of the form y = f(x) + 1. • Outside the HVertical Movement • Since 1 is added to the parent function y = x2,the graph of the parent function moves up 1 unit. a.

18. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = (x - 2)2 • Inside the HHorizontal Movement • This function is of the form y = f(x - 2). • Since 2 is being subtracted from x before being evaluated by the parent function, the graph of the parent function y = x2 slides 2 units right. a.

19. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = (x + 1)2 – 2 • This function is of the form y = f(x + 1)2 -2. The addition of 1 indicates a slide 1 unit left, and the subtraction of 2 moves the parent function y = x2 down two units. a.

20. EXAMPLESmake table and graph

21. Graphs of Nonlinear Inequalities Section 3.3

22. Determine which are solutions • Determine whether (3, 4), (11, 2), (6, 5) and (18, -1) are solutions for the inequality y ≥√x-2) + 3. Of these ordered pairs, (3, 4) and (6, 5) and are solutions for y ≥√x-2) + 3.

23. Example • Determine whether (-2,5) (3,-1) (-4,2) and (-1,-1) are solutions for the inequality y ≥ 2x3+7 • (-2,5) and (-4,2) are solutions

24. Graph y ≤ (x - 2)2 + 2. Since the boundary is included in the inequality, the graph is drawn as a solid curve. To verify numerically, you can test a point not in the boundary. It is common to test (0, 0) whenever it is not on the boundary

25. The boundary is notincluded, so draw it as a dashed line Verify by substituting (0, 0) in the inequality to obtain 0 < -3. Since this statement is false, the part of the graph containing (0, 0) should not be shaded. Thus, the graphis correct. Graph y < -2 - |x - 1|. y < -2 - |x - 1| y < -|x - 1| - 2

26. Solving Absolute Inequalities • Solve |x + 3| - 4 < 2. • There are two cases that must be solved. In one case, x + 3 is negative, and in the other, x + 3 is positive. The solution set is {x | -9 < x < 3}. {x | -9 < x < 3} is read as “the set of all numbers x such that x is between -9 and 3.

27. Solving Absolute Inequalities • Solve |x -2| - 5 < 4. • -(x-2)-5<4 • (x-2)-5<4 • Solution Set {x |-7<x<11}

28. 3.4 Inverse Functions and Relations

29. Inverse Relations • Two relations are inverse relations if and only if one relation contains the element (b,a) whenever the other relation contains the element (a,b). • F(x) – denotes function • F(x)-1 – denotes inverse • Ex. Graph f(x) = -1/2 lxl + 3 and its inverse

30. Horizontal Line Test • A test used to determine if the inverse of a relation will be a function • If every horizontal line intersects the graph of the relation in at most one point, then the inverse of the relation is a function • Is the inverse of the below relation a function? • Ex. F(x) = 3X + 4 • Yes • Ex. F(x) = (x + 3)2 - 5 • No

31. How do you find the inverse? • To find the inverse of a function, let y= f(x) and interchange x and y. Then solve for y. • Ex. F(x) = (x + 3)2 - 5 • F(x)-1 = -3 + -(x+5) 1/2 • Ex. F(x) = 1/(x)3 • F(x)-1 =1/(x)1/3

32. Inverse Functions • Two functions, f and f-1 are inverse functions if and only if (f 0 f-1)(x) = (f-1 0 f)(x)=x. • Ex. Given F(x) = 4x – 9, find f-1 and verify that f and f-1 are inverse functions. • F(x)-1 = (x+9)/4 • Verify: (f 0 f-1)(x) = (f-1 0 f)(x)=x. • Ex. F(x) = 3x2 + 7 • F(x)-1 =((x-7)/3) ½ • Verify: (f 0 f-1)(x) = (f-1 0 f)(x)=x.

33. Continuity and End Behavior Section 3.5

34. Discontinuous • A function is discontinuous if you can not trace it without lifting your pencil. • Infinite Discontinuity – as the graph of f(x) approaches a given value of x, lf(x)l becomes increasingly greater • Jump Discontinuity – The graph of f(x) stops and then begins again with an open circle at a different range value for a given value of the domain. • Point Discontinuity – When there is a value in the domain for which f(x) is undefined, but the pieces of the graph match up • Everywhere Discontinuous – A function that is impossible to graph in the real number system is said to be everywhere discontinuous

35. Continuous Test • A function is continuous at x = c if it satisfies the following conditions: • The function is defined at c, in order words F(c) exists • The function approaches the same y-value on the left and right sides of x = c; and • The y-value that the function approaches from each side is F(c) • Examples: f(x) = 3x2 +7; x = 1 • F(x) = (x-2)/(x2 -4); x = 2 • F(x) = 1/x if x>1, and x if x <=1; x = 1

36. Continuity on an Interval • A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval. • F(x) = 3x-2 if x >2, 2 – x if x <=2 • For 3<x<5 • For 1<x<3

37. End Behavior • The behavior of f(x) as lxl becomes very large • Describe the end behavior of f(x) = -x3 and g(x) = -x3 + x2 –x + 5 • End Behavior Chart of Polynomial Functions pg. 163

38. Increasing, Decreasing, and Constant Functions • A function f is increasing on an interval I if and only if for every a and b contained in I, f(a) < f(b) whenever a < b. • A function f is decreasing on an interval I if and only if for every a and b contained in I, f(a)> F(b) whenever a < b. • A function f remains constant on an interval I if and only if for every a and b contained in I, f(a) = f(b) whenever a < b

39. Monotonicty • A function is said to be monotonic on an interval I if and only if the function is increasing on I or decreasing on I.

40. Examples • Graph f(x) = 3- (x-5)2 • Graph f(x) = ½ lx+3l – 5 • Graph f(x) = 2x3 + 3x2- 12x+ 3

41. Critical Points and Extreme Section 3.6

42. Critical Points • Points on a graph at which a line drawn tangent to the curve is horizontal or vertical. • Three types of critical points: • Maximum: When the graph is increasing to the left of x = c and decreasing to the right of x = c • Minimum: When the graph of a function is decreasing to the left of x = c and increasing ot the right of x = c • Point of Inflection: a point where the graph changes its curvature.

43. Critical Points P P P Point of Inflection Minimum Maximum

44. Critical Points • Extremum – a minimum or maximum value of a function • Relative Extremum – a point that represents the maximum or minimum for a certain interval • Absolute Maximum – the greatest value that a function assumes over its domain • Relative Maximum – a point that represents the maximum for a certain interval • Absolute Minimum – the least value that a function assumes over its domain • Relative Minimum – a point that represents the minimum for a certain interval

45. Critical Points

46. Examples • Graph the following examples and pick out the critical points • F(x) = 5x3 -10x2 – 20x + 7 • Use trace to find relative min and max, and then use calc functions to find max and min • F(x) = 2x5 -5x4 –10x3 has the critical points x = -1, 0, and 3. Determine whether each of these critical points is the location of a max, min, or point of inflection. • Solve by graphing or algebraically

47. Review for Test • Section 3.4 Inverse Relations and Functions • Graph Inverse based on relation graph • Find Inverse • Verify that the Inverse is correct • Section 3.5 Continuity and End Behavior • Describe continuity and the end behavior of a function by graphing it • Section 3.6 • Locate critical points • Identify the critical points as max, min, or point of inflection

48. Graphs of Rational Functions Section 3.7

49. Rational Function • A rational function is a quotient of two polynomial functions. • F(x) = g(x)/h(x); where h(x) does not equal 0. • Parent rational graph is f(x) = 1/x.

50. Asymptotes • A line that a graph approaches but never intersects • Two main types • Vertical Asymptotes: The line x = a is a vertical asymptote for a function f(x) if f(x) approaches infinity or negative infinity as x approaches a from either the left or the right. • Horizontal Asymptotes: The line y=b is a horizontal asymptote asymptote for a function f(x) if f(x) approaches b as x approaches infinity or negative infinity. • Examples: f(x) = (3x-1)/(x-2) • Vertical Asymptotes – look at domain • Horizontal Asymptotes, solve for X or divide both top and bottom by highest power of X.