1. 18 Days

2.  2.1 Definition of a Function 2.1 Definition of a Function  2.2 Graphs of Functions 2.2 Graphs of Functions  2.3 Quadratic Functions 2.3 Quadratic Functions  2.4 Operations on Functions 2.4 Operations on Functions  2.5 Inverse Functions 2.5 Inverse Functions  2.6 Variation 2.6 Variation

3. Two Days

4.  For the function a) Find f(a) b) Find f(a-1) c) Find d) Find

5. p 148 (# 5,8,14-28 even, 45,47,49,52,57)

6.  Day 2 Def Function, Domain, Range, Increasing/Decreasing, Vert Line Test, Def Linear Function Evaluating (p148 #13)

7.  A function F from a set D to a set E is a correspondence that assigns each element x of D to exactly one element y in E.

8.  Domain – The set D is the domain of the function. Domain is the set of all possible inputs.  Range – The set E is the range of the function. Range is the set of all possible outputs.  The element y in E is the value of f at x also called the image of x under f.

9.  We say that f maps D into E.  Two functions f and g are equal if and only if f(x) = g(x) for all x in D.

10.  The graph of a function f is the graph of the equation y = f(x) for all x in the domain of f.  The vertical line test can be used to determine if a graph represents a function. ◦ What does the vertical line test represent in terms of a function mapping?

11. -f is increasing when f(a)<f(b) and a<b. -f is decreasing when f(b)>f(c) and b<c. -f is constant when f(x)=f(y) for all x and y.

12.  Given  Determine the domain of g.  Evaluate g(-3)  Evaluate

13.  What is the difference between sketching and graphing a function?  Why would we sketch a function as opposed to graph a function?

14.  Sketch the following functions and determine the domain, range, and intervals of decreasing, increasing, and constant value:

15.  We can find linear functions in the same way that we find the equation of a line.  If f is a linear function such that f(-3)=6 and f(2)=-12, find f(x) where x is any real number.

16.  Pg 150 #57, 59

17. p 148 (# 15,32,34,35,46,48,50,53,54,60,63,65)

18. Four Days

19. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy -2 00 11 22

20. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy -22 1 00 11 22

21. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy -24 1 00 11 24

22. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy -2-8 00 11 28

23. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy 00 11 42 93 164

24. -Name of Family -Parent Equation -General Equation -Locator Point -Domain-Range xy -8-2 00 11 82

25.  Parent:  Shift up k units:  Shift down k units:  Shift right h units:  Shift left h units  Combined Shift: ◦ (right h units, up k units)

26.  Parent:  Reflection in x-axis:  Vertical Stretch a>1  Vertical Shrink 0<a<1  Horizontal Stretch 0<c<1 :  Horizontal Compression c>1:  Combined Transformation:

27.  Graph the following using translations:

28. Shifts and Reflections WS

29.  Day 2 – Even and Odd functions. Vertical and Horizontal stretching and compressing of graphs.

30.  f is an even function if f(-x)=f(x) for all x in the domain. ◦ Even functions have symmetry with respect to the y-axis. ◦ Ex:  f is an odd function if f(-x)=-f(x) for all x in the domain. ◦ Odd functions have symmetry with respect to the origin. ◦ Ex:

31.  A parent function is the simplest function in a family of certain characteristics.  A translation shifts the graph horizontally, vertically, or both. Resulting in a graph of the same shape in a different location.  A reflection over the x-axis changes y-values to their opposites.

32.  A vertical stretch multiplies all y-values by the same factor greater than 1.  A vertical shrink reduces all y-values by the same factor between 0 and 1.  Each member of a family of functions is a transformation, or change, of the parent function.  A horizontal compression divides all x-values by the same factor greater than 1.  A horizontal stretch divides all x-values by the same factor between 0 and 1.

33.  Parent:  Shift up k units:  Shift down k units:  Shift right h units:  Shift left h units  Combined Shift: ◦ (right h units, up k units)

34.  Parent:  Reflection in x-axis:  Vertical Stretch a>1  Vertical Shrink 0<a<1  Horizontal Stretch 0<c<1 :  Horizontal Compression c>1:  Combined Transformation:

35. pg 164 (# 2,3,5,7,8,13,15,17,20,31-36,39 a- f, 41,42,45)

36.  Day 3 – Piecewise functions and questions from the previous 2 days. Application of Piecewise functions (pg 168 #66)

37.  Piecewise functions are defined by more than one expression over different intervals.  Absolute Value is actually a piecewise defined function.

38.  Lets graph the following piecewise defined function.

40.  An electric company charges its customers $0.0577 per kWh for the first 1000kWh, $0.0532 for the next 4000kWh, and $0.0511 for any over 5000kWh. Write a piecewise defined function C for a customer’s bill of x kWhs.  How much will a customer’s bill be if they used 4300kWh of electricity?

41. pg 167 (# 47-50,53,54,55,56,63-65)

42.  Day 4 – Graphing Piecewise functions WS. Working day for students.

43. Graphing Piecewise Functions WS

44. Two Days

45.  Day 1 – Standard form of a quadratic. Vertex form of a quadratic. Completing the square. Finding x and y intercepts.

46.  Standard form of a Quadratic:  Vertex form of a Quadratic:

48. ● To find the x-intercept, set y=0. Solve for x. ● To find the y-intercept, set x=0. Solve for y.  Find the x and y intercepts of the following:

49. Vertex and Intercepts WS

50.  Day 2 – Vertex formula and Theorem on Max/Min Values.

51.  Standard Form of a Quadratic  Factored Form of a Quadratic

53.  Find the vertex of the following and determine if it is a max or a min:

54.  The length of a frog’s leap is 9ft and has a maximum height of 3ft off the ground. Assuming the frog’s path through the air is parabolic, find an equation that describes the path of the frog through the air.

55. pg 179 (# 1,6,8,10,13,17,20,23,26,27,30,39) pg 179 (# 7,15,31,38,40,43,44,49,54)

56. One Day

57.  We can perform several operation on functions just as we perform the same operation on real numbers. Consider f(x) and g(x):

58.  The composite function of two functions f and g is defined by:  The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.  Essentially, the range of g(x) is the domain of f(x) minus any possible restrictions.

59.  Let and  Find:

60.  Is ????  Consider

61. pg 192 (# 2,3,6,7,9,12,15,21,25,29,30,35,36,39,45,46)

62.  Graph each function below a)Exact x and y intercepts b)Give the domain/range c)Intervals of increase, decrease, constant 1) 2) 3) 4)

63. Two Days

64.  f(x) and g(x) are inverse functions that “undo” one another if and only if.  Notation: Original Function: Inverse Function: Inverses are NOT reciprocals!!

65.  Inverses switch the x and y values of a function. (x,y) -> (y,x)  The domain of f is the range of its inverse.  The range of f is the domain of its inverse.  Graphically, the inverse of a function is a reflection over y=x.

66.  Can you come up with a function that when reflected over the line y=x will no longer be a function?

67.  In order for a function to have an inverse, a fucntion must be 1 to 1. That is, no two elements in the domain can have the same y value.  Ex: is not a 1-1 function.  We can however restrict the domain to find partial inverses. i.e has an inverse.

68.  To find an inverse: ◦ 1. Check if the function is 1-1. Restrict the domain if the original function is not 1-1. ◦ 2. Write f(x) as y. ◦ 3. Switch all x and y in the equation. ◦ 4. Solve for the “new” y. ◦ 5. Rewrite the domain if necessary. ◦ 6. Check that or graph on the TI and check for symmetry about y=x.

69.  Find the inverses to the following functions:

70. pg 203 (# 4,5,7,9-12,14,15,19,23,29,31,35)

71. Finding Inverses WS (# 1-7)

72. One Day

73.  A variation or proportion is used to describe relationships between variable quantities.  k is a nonzero real number called a constant of variation or constant of proportionality.

74.  Direct Variation ◦ y varies directly with x ◦ y is directly proportional to x  Inverse Variation ◦ y varies inversely with x ◦ y is inversely proportional to x  Combined or Joint Variation ◦ z varies jointly with x and y ◦ z varies directly with x and inversely with y

75.  V varies jointly as B and H  P varies directly as the square of V and inversely as R  The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure, P  The distance, D, that a free-falling object falls varies directly as the square of the time, T, that it falls

76.  1. Write the general formula that variables and a constant of variation.  2. Substitute the initial conditions for the variables and solve for the constant of variation k.  3. Substitute the constant of variation k into the general formula from your first step.  4. Use you general formula to solve the problem.

77.  The price, P, of a diamond is directly proportional to the square of the weight, W. If a 1 carat diamond costs $2000, find the price of a 0.7 carat diamond.

78.  The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d. A wire 100ft long of diameter 0.01in has a resistance of 25ohms. Find the resistance of a wire with a diameter of 0.015in and 50ft of length.

79.  pg 209 (# 2-4,6,8,10,13,14,16,17,20,21,23)