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College Algebra

College Algebra

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College Algebra

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  1. College Algebra Exam 3 Material

  2. Ordered Pair • Consists of: • Two real numbers, • Listed in a specific order, • Separated by a comma, • Enclosed within parentheses • Examples of different ordered pairs: (7, 2) (2, 7) (3, -4)

  3. Coordinates of an Ordered Pair • The first number within an ordered pair is called the “x-coordinate” • The second number within an ordered pair is called the “y-coordinate” • Example, given the ordered pair (5, -2) • The x-coordinate is: 5 • The y-coordinate is: -2

  4. Ordered Pairs & Interval Notation • (2, 7) can have two meanings: • It can be an ordered pair. • It can be interval notation and have the meaning of all real numbers between 2 and 7, but not including 2 and 7. • Context will tell the difference, just like in the meaning of certain words. • English Example, “bow”: • ribbon decoration on a package • an instrument for shooting an arrow

  5. Rectangular Coordinate System • Consists of: • Horizontal number line called “x-axis” • Vertical number line called “y-axis” • Intersecting at the zero points of each axis in a point , designated as the ordered pair (0, 0), and called the “origin” • Forms a rectangular grid that can be used to show ordered pairs as points

  6. Graphing Ordered Pair as a Point • The x-coordinate gives horizontal directions from the origin • The y-coordinate gives vertical directions from the origin • When both directions have been followed, the point is located. • Example, graph: (-2, 3)

  7. Rectangular Coordinate System • Axes divide plane into four quadrants numbered counterclockwise from upper right • The signs of the coordinates of a point are determined by its quadrant • I : • II : • III : • IV :

  8. Distance Formula • The “distance” between two points (x1, y1) and (x2, y2) in a rectangular coordinate system is given by the formula: Example: Distance between (2, -1) and (-3, 4)? x1 = 2, y1 = -1, x2 = -3, y2 = 4

  9. Midpoint Formula • The “midpoint” of two points is a point exactly half way between the two points • The formula for finding the coordinates of the midpoint of two points (x1, y1) and (x2, y2) is: • Example: Midpoint of (2, -1) and (-3, 4)?

  10. Homework Problems • Section: 2.1 • Page: 192 • Problems: All: 5 – 14 • MyMathLab Assignment 35 for practice

  11. Ordered Pairs as Solutions to Equations in Two Variables • An equation in two variables, “x” and “y,” requires a pair of numbers, one for “x” and one for “y”, to form a solution • Example: Given the equation: 2x + y = 5 One solution consists of the pair: x = 2 and y = 1 If it is agreed that “x” will be listed first and “y” second, this solution can be shown as the ordered pair: (2, 1)

  12. Ordered Pairs as Solutions to Equations in Two Variables Given: 2x + y = 5 Complete the missing coordinate to form ordered pair solutions: ( 1, __ ) ( 3, __ ) ( __, 5 ) ( __, -3) How many ordered pair solutions can be found?

  13. Graphs ofEquations in Two Variables • A “graph” of an equation in two variables is a “picture” in a rectangular coordinate system of all solutions to the equation • A graph can be constructed by “point plotting,” finding enough (x, y) pairs to establish a pattern of points and then connecting the points with a smooth curve • If the pattern indicates that the curve should extend beyond the graph grid, an arrow is placed on the curve where it exits the grid

  14. Graph: 2x + y = 5 • Find (x, y) pairs by picking a number for “x” and solving for “y”, or vice versa • Show solutions as (x, y) pairs • Plot each point • When pattern is seen connect with smooth curve with arrows at both ends • Solutions:

  15. Graph: • Find and plot solutions (using a “T-chart” helps):

  16. Homework Problems • Section: 2.1 • Page: 194 • Problems: All: 33 – 44 • MyMathLab Assignment 36 for practice

  17. Graphs vs. Form of Equation • Experience will teach us that the general shape of a graphdepends on the form of the equation in two variables • Examples related to last section: • Equations of form: Ax + By = C will always have a straight line graph • Equation of form: y = |mx + b| will always have a V-shaped graph

  18. Center Radius Equation of Circle • Equation of the form: • Will always have a graph that is a circle with center: • and radius:

  19. Center Radius Equation • Considering: • What would be the equation of a circle with center (-1, 2) and radius 3?

  20. Center Radius Equation • Given each of the following center radius equations, find the center and the radius:

  21. Homework Problems • Section: 2.1 • Page: 194 • Problems: All: 45 – 52 • MyMathLab Assignment 37 for practice

  22. General Equation of a Circle • The equation will represent a real circle, if, when put in “center-radius form”, Example: • Don’t know yet, if it is a “real” circle, until written:

  23. Converting General Equation to Center-Radius Equation • “Complete the Square” twice, once on “x” and once on “y” (keep both sides balanced) Example: • The “general equation” is a “real” circle since: • Center and radius:

  24. Homework Problems • Section: 2.1 • Page: 195 • Problems: All: 57 – 64 • MyMathLab Assignment 38 for practice • MyMathLab Quiz 2.1 due for a grade on the date of our next class meeting

  25. Relation • Relation – any set of ordered pairs Example: M = {(-3,2), (1,0), (4,-5)} • Domain of a Relation – set of first members (“x coordinates”) Example: Domain M = {-3, 1, 4} • Range of a Relation – set of all second members (“y coordinates”) Example: Range M = {2, 0, -5}

  26. Equations in Two Variables • Considered to be relations because solutions form a set of ordered pairs Example: • There are an infinite number of ordered pair solutions, but some are:

  27. Domain ofEquations in Two Variables • Set of all x’s for which y is a real number • May help to find domain by first solving for “y” Example: • Easy to see that x can be anything except -1: Domain =

  28. Range ofEquations in Two Variables • Set of all y’s for which x is a real number • May help to find range by first solving for “x” Example: • Easy to see that y can be anything except 3: Range =

  29. Function • A special relation in which each x coordinate is paired with exactly one y coordinate Example – only one of these is a function: R = {(2,1), (3,-5), (2,3)} Not a Function S = {(3,2), (1,2), (-5,3)} FUNCTION

  30. Homework Problems • Section: 2.2 • Page: 209 • Problems: All: 5 – 15 • MyMathLab Assignment 39 for practice

  31. Dependent & Independent Variables in Functions • Since every x is paired with exactly one y, we say that “y depends on x” • For a function defined by an equation in two variables, x is the “independent variable” and y is the “dependent variable”

  32. Functions Definedby Equations in Two Variables • To determine if an equation in two variables is a function, solve for y and consider whether one xcan give more than one y – if not, it is a function Example – only one of these is a function: • Solve each equation for y and analyze:

  33. Example Continued

  34. Function Notation • Functions represented by equations in two variables are traditionally solved for ybecause doing so shows how y depends on x Example – each of these is a different function:

  35. Function Notation Continued • When working with functions it is also traditional to replace y with the symbol f(x) or some variation using a letter other than f. This gives different functions different names: Previous Example Written in “Function Notation”

  36. Function Notation Continued • Function notation “f(x)” is read as “f of x” and means “the value of y for the given x” Example: If: Then: This means that for this f function, when x is -2, y is -7

  37. Graphs ofRelations and Functions • Can be accomplished by point plotting methods already discussed • The graph of a relation will be a function if, and only if, every vertical line intersects the graph in at most one point (VERTICAL LINE TEST)

  38. Example of Vertical Line TestWhich graph represents a function? • Above: passes vertical line test – Function • Below: fails vertical line test – Not a Function

  39. Practice Using Function Notation • Given the functions f, g and h defined as follows, and remembering that f(2) means “the value of y in the f function when x is 2”, find the value of each function for the value of “x” shown: If If g = {(1,3),(-2,4),(2,5)}findg(-2) g(-2) = 4

  40. Practice Using Function Notation Given the graph of y = h(x), find h(-3) h(-3) = 1

  41. Determining if Equation in Two Variables is a Function • Solve the equation for y. • If one x gives exactly one y, it is a function. • If it is a function, it may be written in function notation by replacing y with f(x).

  42. Example Determine if the following equation defines a function, if so, write in function notation: This is a function since one x gives one y

  43. Increasing, Decreasing and Constant Functions • A function is increasing over some interval of its domain if its graph goes up as x values go from left to right within the interval • A function is decreasing over some interval of its domain if its graph goes down as x values go from left to right within the interval • A function is constant over some interval of its domain if its graph is flat as x values go from left to right within the interval

  44. Example • Show the “interval of the domain” where the given function, • is increasing: • is decreasing: • is constant:

  45. Homework Problems • Section: 2.2 • Page: 210 • Problems: Odd: 17 – 37, 41 – 63, 69 - 77 • MyMathLab Assignment 40 for practice • MyMathLab Quiz 2.2 due for a grade on the date of our next class meeting

  46. Linear Functions • Any function that can be written in the form: Example: f(x) = 3x - 1 • This is called the slope intercept form of a linear function (reason for name – explained later). • The graph of a linear function will always be a non-vertical straight line

  47. Graphing Linear Functions • Find any two ordered pairs that are solutions (chose two different x’s and find corresponding y’s) • Plot the two points in a rectangular coordinate system. • Draw a straight line connecting the two points with arrows on both ends.

  48. Graphing a Linear Function Graph f(x) = 3x – 1 Choose two values of “x” and calculate corresponding “y” values: f(0) = 3(0) – 1 = -1 [ordered pair: (0,-1)] f(2) = 3(2) – 1 = 5 [ordered pair: (2, 5)]

  49. Linear FunctionsWith Horizontal Line Graphs A linear function of the form: can be simplified to: This means y will always be the same (y = b) no matter the value of “x” The graph will be a horizontal line through all the points that have a “y” value of “b”

  50. Example of Linear FunctionWith Horizontal Line Graph Graph f(x) = -3 (Note: this function says no matter what value “x” is, the “y” value will be -3.)