The document describes five main families of functions - linear, power, root, reciprocal, and absolute value functions. It provides the name, equation, domain and range for each type of function. It also discusses concepts like piecewise functions, average rate of change, transformations, combinations of functions, and variations.

1. AFM – CHAPTER 4 Functions

2. 5 Function Families What you need to know: Name Equation Domain Range

3. Linear Name – Constant Equation – Domain – (-  ,  ) Range – [b]

4. Linear Name – Oblique Linear Equation – Domain – (-  ,  ) Range – (-  ,  )

5. Power Functions Name – Quadratic Equation – Domain – (-  ,  ) Range – [0,  )

6. Power Functions Name – Cubic Equation – Domain – (-  ,  ) Range – (-  ,  )

7. Root Functions Name – Square root Equation – Domain – [0,  ) Range – [0,  )

8. Root Functions Name – Cube root Equation – Domain – (-  ,  ) Range – (-  ,  )

9. Reciprocal Functions Name – Rational Equation – Domain –(-  ,0)  (0,  ) Range – (-  ,0)  (0,  )

10. Reciprocal Functions Name – Rational Squared Equation – Domain – (-  ,0)  (0,  ) Range – (0,  )

11. Absolute Value Function Name – Absolute value Equation – Domain – (-  ,  ) Range – [0,  )

12. Greatest Integer Function Name – Greatest Integer Equation – Domain – (-  ,  ) Range – (integers)

13. Trig Functions Name – Sine Equation – Domain – (-  ,  ) Range – [-1,1]

14. Trig Functions Name – Cosine Equation – Domain – (-  ,  ) Range – [-1,1]

15. Vertical Line Test A curve in the coordinate plane is the graph of a function iff no vertical line intersects the curve more than once.

16. Piecewise Functions Sketch the graph of:

17. Equations That Define Functions Does the equation define y as a function of x ?

18. Homework 4.2 Page 228 – 230; 2 – 50 even 60 –70 even Know Family of Functions graphs Name, equation, domain, range

19. 4.4 – Average Rate of Change Average rate of change of a function f(x) between x = a and x = b is: Slope of secant line drawn between x=a & x=b or line through the points (a,f(a)),(b,f(b)).

20. Finding the Average Rate of Change Find the average rate of change if

21. Example 2 An object is dropped from a height of 3000 feet, its distance above the ground h , after t seconds is given by: Find the average speed Between 1 & 2 seconds Between 4 & 5 seconds

22. Increasing and Decreasing Functions A function f is increasing if: A function f is decreasing if:

23. State the intervals on which the function whose graph is shown is increasing or decreasing.

24. Sketch the graph of the function: Find the domain and range of the function Find the intervals on which f increases and decreases

25. Determine the average rate of change of the function between the given values of the variable. Use a graphing calculator to draw the graph of f . State approximately the intervals on which f is increasing and on which f is decreasing.

26. The table gives the population in a small coastal community for the period 1990 – 1999. Figures shown are for January 1 in each year. What was the average rate of change of population between 1991 and 1994? What was the average rate of change of population between 1995 and 1997? For what period of time was the population increasing? For what period of time was the population decreasing?

27. Transformations Vertical Shift Horizontal Shift Reflecting Stretching/Shrinking

28. Exploring transformations Graph Graph

29. More transformations Graph: Graph:

30. General Rules for Transformations Vertical shift: y=f(x) + c  c units up y=f(x) – c  c units down Horizontal shift: y=f(x+c)  c units left y=f(x-c)  c units right Reflection: y= – f(x)  reflect over x -axis y= f(-x)  reflect over y -axis Stretch/Shrink: y=af(x)  (a > 1) Stretch vertically y=af(x)  (0 < a < 1) Shrink vertically

31. Even & Odd Functions Algebraically: Even – f is even if f(-x) = f(x) Odd – f is odd if f(-x) = - f(x) Graphically: Even – f is even if its graph is symmetric to the y -axis Odd – f is odd if its graph is symmetric to the origin

32. Determine Algebraically if the function is even, odd or neither

33. Use the rules of transformations to graph the following:

34. 4.7 Combining Functions Combining – Addition, Subtraction, Multiplication, or Division Composition of functions – Putting two functions together using the rules of one of the functions

35. Combining Functions Addition/Subtraction – f(x) and g(x) (f ± g)(x) = f(x) ± g(x)  Add/Subtract, then combine like terms Domain: D:f(x)  D:g(x) Multiplication – f(x) and g(x) (fg)(x) = f(x) ·g(x)  Multiply, then combine like terms Domain: D:f(x)  D:g(x) Division – f(x) and g(x)  Divide, then simplify Domain: D:f(x)  D:g(x), where g(x)  0

36. Examples Let Find Domain f(x)  Domain g(x)  Find

37. Composition of Functions

38. Examples If : and Find:

39. Composition of 3 Functions Find: If: Page 276 - # 23,25,27

40. Variation Direct Variation Indirect Variation Joint Variation

41. Direct Variation y varies directly as x y is directly proportional to x y is proportional to x Formula (Equation to use) y = kx  ;k is constant of proportionality

42. During a thunderstorm, the distance between you and the storm varies directly as the time interval between the lightening and thunder. Suppose thunder from a storm 5400 ft away takes 5 seconds reach you. Determine the constant of proportionality and write the variation equation for the model. Sketch the graph. What does the k represent? If the time interval between the lightening and thunder is 8 sec. How far away is the storm?

43. Inverse Variation y varies inversely as x y is inversely proportional to x Formula (Equation to use) 

44. Boyle’s Law – When a sample of gas is compressed at a k onstant temperature the pressure of the gas is inversely proportional to the volume of the gas. P – pressure v – volume k – constant of proportionality Suppose the pressure of a sample of air that occupies 0.106 m ³ @ 25ºC is 50 kPa. Find the constant of proportionality and write the equation that expresses the inverse proportionality If the sample expands to a volume of .3m³, find the new pressure

45. Joint Variation Used when a quantity depends on more than one other quantity.  It depends on them jointly. z varies jointly as x and y z is jointly proportional to x and y z is proportional to x and inversely proportional to y

46. Newton’s Law of Gravitation – Two objects with masses m 1 and m 2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express Newton’s Law of Gravitation as an equation.

47. Examples of Variation Write an equation that expresses the statement: R varies directly as t . v is inversely proportional to z . y is proportional to s and inversely proportional to t . R is proportional to j and inversely proportional to the squares of s and t .

48. Express the statement as a formula. Use the given information to find the constant of proportionality y is directly proportional to x . If x = 4, then y = 72. M varies directly as x and inversely as y . If x = 2 and y = 6, then M = 5. s is inversely proportional to the square root of t . If s = 100, then t = 25.

49. Hooke’s Law states that the force F needed to keep a spring stretched x units beyond its natural length is directly proportional to x . Here the constant of proportionality is called the spring constant. Write Hooke’s Law as an equation. If a spring has a natural length of 10 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant. What force is needed to keep the spring stretched to a length of 14 cm?

50. The resistance R of wire varies directly as its length L and inversely as the square of its diameter d . Write an equation that expresses this joint variation. Find the constant of proportionality if a wire 1.2 m long and 0.005 m in diameter has a resistance of 140 ohms. Find the resistance of a wire made of the same material that is 3 m long and has a diameter of 0.008 m.

51. Modeling Quadratic & Cubic Functions Define the variable Find the equation (model) Answer the question(s) asked

52. A breakfast cereal company manufactures boxes to package their product. The prototype box has the following shape: Its width is three times its depth and its height is five times its depth. Find a function that models the volume of the box in terms of its depth.

53. A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $14, average attendance at recent games has been 9500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000. What price maximizes revenue from ticket sales, and what is the maximum revenue?

54. A manufacturer makes a metal can that holds 1 L(liter) of oil. What radius minimizes the amount of metal in the can?

55. A gardener has 140 feet of fencing for her rectangular vegetable garden. Find the dimensions of the biggest area she can fence.