Algebra II - Segments Lines and Inequalities | Mathematics

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Study GuideAlgebra II–Segments Lines and InequaliƟes1. Rectangular Coordinate SystemArectangular coordinate systemhelps us locate points on a graph using numbers. Every point on agraph is described bytwo coordinates, written as anordered pair:(x, y).These coordinates are also calledCartesian coordinates, named after the French mathematicianRené Descartes, who developed this system.When we know:•theslopeandinterceptof a line, or•thecoordinates of two pointson the linewe can determine theequation of that line.Key Terms You Should KnowLet’s go through some important terms that are used when working with coordinate systems.1. Coordinates of a PointOn anumber line, each point corresponds to a single number.Similarly, on acoordinate plane, each point is identified bytwo numbers.These two numbers together are called thecoordinates of the point.Example:The point(3, 4)means:•move3 units along the x-direction•then4 units along the y-direction

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Study Guide2. x-axis and y-axisTo locate points on a plane, we usetwo perpendicular lines:•Thex-axis→ahorizontal line•They-axis→avertical lineThese two axes intersect at the center of the plane.3. Coordinate PlaneThecoordinate planeis the entire flat surface that includes:•thex-axis•they-axis•all the pointson the graphIt is the system we use to plot and locate points.4. Ordered PairsEvery point on a coordinate plane is written as anordered pair:Theorder matters.•Thefirst number (x)tells how far to moveleft or right.•Thesecond number (y)tells how far to moveup or down.Example:(2, 5)means:•move2 units right•move5 units up

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Study Guide5. OriginThe point where thex-axis and y-axis intersectis called theorigin.The coordinates of the origin are:It is thestarting pointfor locating all other points.6. x-CoordinateThex-coordinateis thefirst numberin an ordered pair.It shows movementparallel to the x-axis.•Positive x-value→moveright•Negative x-value→moveleftExample:•(4, 2)→move4 units right•(-3, 2)→move3 units left7. y-CoordinateThey-coordinateis thesecond numberin an ordered pair.It shows movementperpendicular to the x-axis.•Positive y-value→moveup•Negative y-value→movedownExample:•(2, 5)→move5 units up•(2,-4)→move4 units down

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Study Guide8. QuadrantsThex-axis and y-axis divide the coordinate plane into four regionscalledquadrants.Figure 1. The quadrants.The quadrants are numberedcounterclockwise, starting from the upper-right.Quadrant I•x is positive•y is positiveQuadrant II•x is negative•y is positive

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Study GuideQuadrant III•x is negative•y is negativeQuadrant IV•x is positive•y is negativeGraph of an Ordered PairWhen we place a point on the coordinate plane using its ordered pair, that plotted point is called thegraph of the ordered pair.For example:•The graph of(2,3)is the point located2 units right and 3 units upfrom the origin.Summary•Arectangular coordinate systemis used to locate points on a graph usingordered pairs(x, y).•Thex-axisis horizontal and they-axisis vertical.•Their intersection point is called theorigin (0,0).•Thex-coordinatetells how far to moveleft or right.•They-coordinatetells how far to moveup or down.•The coordinate plane is divided intofour quadrants:

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Study GuideoQuadrant I→(+,+)oQuadrant II→(-,+)oQuadrant III→(-,-)oQuadrant IV→(+,-)•Plotting an ordered pair on the plane gives thegraph of that point.2. Quiz: Rectangular Coordinate System1. In which quadrant is the point with coordinates (–3,–21)?Answer Choices•quadrant II•quadrant III•quadrant IVCorrect Answerquadrant IIIWhy This Is CorrectThe quadrant of a point depends on thesigns of the x-and y-coordinates.•Ifx is negativeandy is negative, the point lies inQuadrant III.For the point(–3,–21):•x =–3 (negative)•y =–21 (negative)Sinceboth coordinates are negative, the point is located inQuadrant III.

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Study Guide2. In which quadrant is the point with coordinates (21,–3)?Answer Choices•quadrant II•quadrant III•quadrant IVCorrect Answerquadrant IVWhy This Is CorrectTo find the quadrant, look at thesigns of x and y.•Ifx is positiveandy is negative, the point lies inQuadrant IV.For(21,–3):•x = 21 (positive)•y =–3 (negative)Apositive xandnegative yplace the point inQuadrant IV.3. Which describes the order of the signs of an ordered pair in the second quadrant?Answer Choices•positive, negative•negative, positive•positive, positiveCorrect Answernegative, positive

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Study GuideWhy This Is CorrectIn thesecond quadrant, the coordinates follow this pattern:•x is negative•y is positiveSo ordered pairs in Quadrant II look like(–, +).Example:(–4, 7)lies in Quadrant II because the x-coordinate is negative and the y-coordinate ispositive.4. Which describes the order of the signs of an ordered pair in the fourth quadrant?Answer Choices•positive, negative•negative, positive•positive, positiveCorrect Answerpositive, negativeWhy This Is CorrectIn thefourth quadrant, the coordinates follow this pattern:•x is positive•y is negativeSo ordered pairs in Quadrant IV look like(+,–).Example:(5,–2)is in Quadrant IV because x is positive and y is negative.

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Study Guide5. Which describes the order of the signs of an ordered pair in the third quadrant?Answer Choices•positive, negative•negative, positive•negative, negativeCorrect Answernegative, negativeWhy This Is CorrectIn thethird quadrant, both coordinates are negative.•x is negative•y is negativeSo ordered pairs in Quadrant III look like(–,–).Example:(–6,–4)lies in Quadrant III because both coordinates are negative.3. Distance FormulaIn coordinate geometry, we often need to find thedistance between two pointson a graph. Thedistance formulahelps us calculate this distance using the coordinates of the points.To understand where this formula comes from, let’s look at an example.Example with Three Points

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Study GuideFigure 1. A right triangle.Suppose we have three points on a coordinate plane:•Point A = (2, 2)•Point B = (5, 2)•Point C = (5, 6)These points form aright triangle.Finding ABTo find the length ofAB, we only need to look at thex-coordinatesbecause both points have thesamey-value (2).So, the length ofAB is 3 units.

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