1. College Algebra Acosta/Karwoski

2. CHAPTER 1 linear equations/functions

3. CHAPTER 1 Sections 1 and 2

4. PRELIMINARY VOCABLULARY

5. Vocabulary Expression vs equation equivalent form Evaluate, simplify, solve, factor Information vs question Process vs answer number, line, point, interval

6. Linear equations Definition : Equations that have ONLY multiplication and addition with degree 1 variables and terms.

7. Linear equations

8. Change the form of a linear equation Solve linear equations Finding points – general, specific, intercepts Slope of line/ rate of change Significance of information: ie word problems Linear graphs

9. SOLVING LINEAR EQUATIONS General solutions, intercepts

10. Solving- finding a value for the variable(s) Algebraically One variable: Two variable specific general intercepts Forms of equations y = mx + b Ax +By + C = 0 (y – y 1 ) = m(x – x 1 ) From graph- entering equations – window settingw [xmin, xmax,xscale] x [ymin, ymax, yscale] From table

11. Solving algebraically for one variable There are 2 numbers in an equation that are equal Changing one of them changes the truth of the equation – changing both of them the same way changes the form of the equation but not the truth of the equation Isolate the variable – changing the equation so that the variable is on one side (alone), makes a statement about the value of x that is true.

12. Examples: solving for one variable

13. Solving for 2 variables 2 variables in an equation form a relation between the two variables A solution to the equation is a pair of numbers For any 2 variable equation there are infinite solutions

14. Specific points x48 y-312

15. General solutions For 3x + 2y = 9 find 5 solutions For y = 3x/5 – 2 find 3 solutions

16. Intercepts An intercept is a point- a solution pair where a graph crosses the x- axis or the y- axis. ( x-intercepts are frequently called roots ) An x-intercept (root) has a y-coordinate of zero A y-intercept has an x- coordinate of zero Thus intercepts are also sometimes referred to as the zeroes of the equation

17. Finding intercepts : algebraically.2x -.6y = 2.4 y = 3x – 7 3000 + y = 6x

18. Using a graph to check solutions Setting the graphing window Entering a graph Using the trace key

19. Examples: check solutions

20. Using a table Enter equation 3(x – 9) + 5x – 4 = 15 Table set Look up Guess and check

21. SLOPE AND RATE OF CHANGE

22. Slope/ rate of change Slope is a measure of the angle of a line Rate of change is a ratio used to compare the change in two numeric items. Slope IS a rate of change – for a line it is constant

23. What you should know about slope the line is increasing if and only if the slope is positive The line is decreasing if and only if the slope is negative If the line is horizontal the slope is 0. If the line is vertical the slope is undefined Slope is a ratio between changes in two different things - in other words slope is a rate – rates MULTIPLY so in y = mx + b m is the slope of the corresponding linear graph

24. slope Given points Given graph Given equation

25. Examples find the slope of lines that pass through the following points (2,9) (-3,8) (-6,3) (3, - 4) ( 0, 6) (5,6) (-5, 4) (-5,8)

26. More examples

27. Rate of change/ slope

28. Examples: find the slope of each line described here

29. Sketching linear graphs

30. APPLICATIONS Significance of information

31. Understanding answers in a real life graph x and y have labels Therefore ordered pairs have labels The label on slope comes from these labels Ex: a graph represents the calories consumed by running over a period of hours. therefore y is calories x is hours The point (2, 4034) means? The point (0,500) would mean? Is this point likely to be a solution point? What about (-3, 268)? What does (4, 0) mean? If the graph is linear and its slope is 5297/2 what does this number mean?

32. Example y =.3x + 32 is the equation for your bill in $ under a phone plan where x is the number of minutes you talked. If you talk for 20 minutes what is your bill? If your bill is $49.10 how many minutes did you talk? What is the y-intercept - what does it mean? What is the slope? What does it mean? What is the x – intercept ? What does it mean? What is an appropriate domain (window) for this problem?

33. CHAPTER 1 – SECTION 3 Functions vs relations

34. Relation A relation is any statement, table, graph, diagram that creates a connection between two sets. the sets are called domain and range any equation that contains 2 variables is a relation Functions are not always straight lines Functions are not always equations

35. Examples: these are all relations y = 3x + 5 x 2 + y 2 = 12 x224-65.16 y-423786

36. Domain and range Sets of numbers Domain – the set of numbers that are reasonable replacements for the x coordinate Range – the set of numbers that correspond to the numbers in the domain thus forming solution pairs. Domain and range are NOT single numbers. The are frequently interval of numbers

37. Examine the domain and range y = 3x + 5 x 2 + y 2 = 12 x224-65.16 y-423786

38. Restricted domains Domains can be restricted by choice (as in choosing a window for the calculator) Domains can be restricted words in an application problem Domains can be restricted by operators The natural domain of a function is all real numbers unless some numbers are restricted by one of the above, in which case the domain is all numbers except those restricted values Domain is an interval

39. Restrictions you should know Fractions cannot have a zero in the denominator - therefore any values that create a zero in the denominator are not in the domain square roots (4 th roots, 6 th roots, etc) cannot have a negative number inside the radical sign – therefore any values that cause the radicand to be negative are restricted

40. Examples

41. Functions A relation with the characteristic that each element in the domain is related to ONLY one value in the range is a function Domain takes on the sense of an input value and range is the output. The relation now is about how the value of (range variable) DEPENDS on the value of (domain variable) Which set is chosen for domain affects whether you have a function or not.

42. Example: decide if the following relations are function a report is created that connects gpa to student number : domain ? range? function?

43. Examples continued: is the relation a function 2x + 2y = 14 y = 3x 2 – 5x NOTE: Any equation that can be solved uniquely for y is a function of x. X123456789 y642101234

44. Function notation The symbols g(x) = y mean: There is a rule (not necessarily an equation) called “g” that has an input of x with a result of y What does h(2) = -4 mean? therefore k(x) = 2x + 5 means?

45. Questions using f(x) notation given what is f(2014)? for what value does k(x) = 92? what is the domain of the function? what is the range of the function? X2012201420162020 K(x)659243120

46. Questions using f(x) notation

47. Using table feature on calculator X 64.832.41 n(x)

48. CHAPTER 1 SECTION 4 Interpreting/reading function graphs

49. Information from graphs - for ANY relation not just linear Do I have a function? Domain - (the interval covered by the graph – left to right) Range – (the interval covered by the graph – down to up) Given input find output ( f(x) = ? - find a point) Given output find input (f(??) = y - find a point Find y- intercepts (a point on y axis) Find x – intercepts (a point on x axis)

50. Is it a function? If output is unique for input – yes “Vertical line test”

51. Domain and range Stated as intervals. Pay attention to closed and open endpoints and arrows. Note any “skipped” sections

52. Given input estimate output f(2) = 3 means that (2,3) is a point on the graph. find f(5) means to find the output when the input is 5. If the graph is a function output is unique but not always defined. Find: k(2) k(-3) k(0) k(4) k(x)

53. Given output estimate input (solve) Find a value for x so that f(x) = 9. means find the input value that gives you 9 There can be MORE than one answer to this question. There can be no solution to this question!!!!! Find x so that: m(x) = 5 m(x) = 2 m(x) = 0 m(x) = -8

54. Y and x intercepts solution points on the axis a function has at MOST one y –intercept but may have 1 x-intercept, no x – intercepts or many x- intercepts

55. New questions – using graphs Where is the function Increasing (intervals) Where is the function Decreasing (intervals) On what intervals is f(x) < # On what intervals is f(x) > #

56. Increasing and decreasing intervals This question is about the domain. Increasing is going up when read from left to right Decreasing is going down when read from left to right

57. f(x) > a f(x)<a The answer to an inequality is always an interval - the answer to these questions is a domain interval draw a horizontal line through a. Where the graph crosses this line is the endpoints of the solution intervals

58. Example Estimate the values for which f(x) < 2 f(x) > 7 f(x) ≤ 0

59. CHAPTER 1 SECTION 5 Linear functions

60. Linear functions are linear equations in 2 variables Linear equation in 2 variables Contains only multiply and add can be written as y = mx +b as long as the equation contains a y variable Always graph a line rate in equation is same as slope of line Given y you can find x Given x you can find y Linear function Has only multiplying and adding ALL linear equations are functions EXCEPT x = constant Is the only function that graphs a straight line Has a constant rate of change ( either increases everywhere or decreases everywhere) Has one y intercept Has at most one x – intercept domain and range are both ALL real numbers - f(x) = y

61. Models - equations A model is a mathematical equation that describes the data for a real life event where there is a pattern that matches that equation Some models are exact – like interest/time models Some models are estimates – like distance/time If there the pattern in the data has a constant rate of change it is linear.

62. Finding equations for linear models a line is described by 2 points It can also be described by one point and its slope If you can describe or picture the line you can determine the equation that generates the line This can be done by many different methods - We will explore 3.

63. y = mx +b method Given the line through (0,2) with slope of 3/5 Given the line through (2,7) with slope of 5 Given the line through (-4,7) and (5,3)

64. Point slope method

65. Parallel and perpendicular lines parallel lines have the same slope Write an equation for a line parallel to y= 3x – 9 going through the point (2,7)

66. Horizontal and vertical lines A horizontal line has slope of 0. A horizontal line has a y intercept --- (0,b) and ALL other points have a y co- ordinate of b So the equation will be y = b A horizontal line through (9, 7) has an equation of? A vertical line has NO slope – is NOT a function therefore does NOT contain a y variable - and has an x intercepts or (x,0) where ALL points have the same x value. So the equation for a vertical line through (9,7) is?