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Linear Functions

Linear Functions. Rates of Change and Systems. Bicycles. In groups of four, generate the function that relates the number of wheels to the number of bicycles. Represent it Graphically, Symbolically, Numerically, Verbally. Post your results on the wall. Definition of a Linear Function.

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Linear Functions

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  1. Linear Functions Rates of Changeand Systems

  2. Bicycles In groups of four, generate the function that relates the number of wheels to the number of bicycles. Represent it Graphically, Symbolically, Numerically, Verbally. Post your results on the wall.

  3. Definition of a Linear Function • What is it that makes a function a linear function? • If you require crutches for a leg injury the function can compute the crutch length in inches for a person x inches tall. Is this a linear function? Why?

  4. Definition of a Linear Function • One defining property that linear functions have is a fixed rate of change. • For example: • Notice that for an increase of 1 in x we always get an increase of 3 in y.

  5. What is The Fixed Rate of Change?

  6. Definition of a Linear Function • Any function that can be represented bywhere mand b are constants, is a linear function.

  7. Example • To enroll at College of the Redwoods you must pay a $10 health fee and $11 per unit (assuming you’re a California resident.) • Notice there is a fixed price, $10, and a variable cost, $11 per unit. Notice that the variable cost is a ratio of dollars to units taken.

  8. Example • So, the linear function would look like the following:where x is the number of units taken. • How much would you need to pay for enrolling in 5 units?, 12 units? What is the domain and range of this function?

  9. Graph of a Linear Function

  10. Ponder • What is the least number of quadrants the graph of a linear function can go through? Give an example. • What is the most number of quadrants a linear function can go through? Give an example.

  11. Linear Functions from Data. Average Cost of College Tuition. • Is this a function? If so, what is the domain and range? Is it discrete? Give a graphical representation.

  12. Average Cost of College (1997,3111) (1996,2966) (1995,2811) (1994,2705)

  13. Symbolic Form of the Function • Based on this data, can we find a symbolic form of the function that relates the year to the cost? • Does it appear that a line would be the best model of the function? • Say we use the first point and the last point.

  14. Average Cost of College (1997,3111) (1996,2966) (1995,2811) (1994,2705)

  15. Continuous Model of a Discrete Function • Notice that for ease of use we have decided to model a discrete function with a continuous one. This is very commonly done but it is rarely explicitly stated.

  16. Examples From Text Books

  17. Rate of change • What is the change in cost for tuition per year based on our two points?

  18. Dollars = $406 Years = 3 Average Cost of College

  19. How To Find A Linear Function Given Two Points

  20. y y2 y1 x x1 x2 Slope of a Linear Function

  21. Slope of a Line

  22. Any stinkingpoint on the line y x Slope Intercept Form y-intercept

  23. Point Slope Form Any stinkingpoint on the line y Any stinking fixedpoint on the line x

  24. Slope Formula Slope Intercept Form Point Slope Form Line Formulas Standard Form

  25. End OfHow To Find A Linear Function Given Two Points

  26. Dollars = $406 Years = 3 Average Cost of College

  27. Quick Note Whenever the domain of a function is the natural numbers, the function is a sequence. Notice the range can be any set. Because a sequence is a function, many of the concepts of functions apply. Instead of letting the convention is to writewhere n is in the natural numbers. This change in notation let’s your readers know that they are now working with a discrete function or a sequence.

  28. Let’s Kix It

  29. Linear Systems • Solve the following system of linear equations graphically:

  30. Graphically

  31. Numerically

  32. Symbolically • Substitution

  33. Symbolically • Elimination

  34. Types of Solutions

  35. Examples

  36. Using Systems to Solve Equations • The graphical method of solving systems of equations can be extended to solving many types of equations. For example:

  37. The Numerical Method

  38. Your Turn • Give all 4 representations of the solution.

  39. y x x1 x2

  40. Solve Linear Equations With Different Notations.

  41. Matrices • We can use matrix equations to solve systems of linear equations in two or more variables. • A rectangular array of numbers is a matrix. Each number in a matrix is called an element.

  42. Matrices

  43. Augmented Matrix

  44. Go Other Way

  45. Gaussian Elimination • We use the elementary row operations to place an augmented matrix in reduced row echelon form. Upon doing so, the solution is easy to read.

  46. Elementary Row Operations • Any two rows can be interchanged • The elements of any row can be multiplied by a nonzero number. • Any row may be changed by adding to its elements a multiple of the corresponding elements of another row.

  47. Example

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