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Algebra I – Coordinate Geometry - Document preview page 1

Algebra I – Coordinate Geometry - Page 1

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Algebra I – Coordinate Geometry

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Algebra I – Coordinate Geometry - Page 1 preview imageStudy GuideAlgebra ICoordinate Geometry1. Coordinate GraphsIntroduction to Coordinate GeometryCoordinate geometry helps usdraw and study points, lines, and shapes on a gridcalled thecoordinate plane(or coordinate graph).You may already know that every point on anumber linerepresents a number.In the same way, every point on acoordinate planeis represented bytwo numbers.These two numbers showexactly where a point is locatedon the graph.To understand how this works, we use two special lines calledcoordinate axes.Thehorizontal lineis called thex-axis.Thevertical lineis called they-axis.These two lines cross each other at a point called theorigin.Theoriginrepresents the coordinate(0, 0).
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Algebra I – Coordinate Geometry - Page 2 preview imageStudy GuideFigure 1An x-y coordinate graphUnderstanding CoordinatesEvery point on the coordinate plane is described using anordered pair of numbers, calledcoordinates.An ordered pair looks like this:[(x, y)]This means:Thefirst number (x)shows thehorizontal position.Thesecond number (y)shows thevertical position.Important Termsx-coordinate (Abscissa)The first number in the ordered pair.It tells how far a point movesleft or right.y-coordinate (Ordinate)The second number in the ordered pair.It tells how far a point movesup or down.ExampleFor the point(1, 4):Move1 unit rightfrom the origin.Then move4 units up.That location marks the point(1, 4)on the graph.
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Algebra I – Coordinate Geometry - Page 3 preview imageStudy GuideFigure 2Graphing or plotting coordinates(1, 4)(0, 2)(6, 2)(3,1)(3, 2)(4,2)Positive and Negative DirectionsOn a coordinate graph, numbers can bepositive or negativedepending on their position.On the x-axisNumbers to theright of 0arepositiveNumbers to theleft of 0arenegativeOn the y-axisNumbersabove 0arepositive
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Algebra I – Coordinate Geometry - Page 4 preview imageStudy GuideNumbersbelow 0arenegativeThis system helps us find the exact location of any point on the graph.Quadrants of the Coordinate PlaneThe coordinate plane is divided intofour sectionscalledquadrants.These quadrants are labeled usingRoman numerals:Quadrant IQuadrant IIQuadrant IIIQuadrant IVEach quadrant contains points with different combinations of positive and negative coordinates.Figure 3Coordinate graph with quadrants labeledPlace this imageright after introducing quadrants.
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Algebra I – Coordinate Geometry - Page 5 preview imageStudy GuideSigns of Coordinates in Each QuadrantRemember these patterns to quickly identify the quadrant of a point.Quadrant Ix ispositivey ispositiveExample: (3, 4)Quadrant IIx isnegativey ispositiveExample: (3, 4)Quadrant IIIx isnegativey isnegativeExample: (3,4)Quadrant IVx ispositivey isnegativeExample: (3,4)A helpful trick students use is:Start in the top-right and move counterclockwise:IIIIIIIVGraphing Equations on the Coordinate Plane
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Algebra I – Coordinate Geometry - Page 6 preview imageStudy GuideSometimes we want tograph an equationon a coordinate plane.To do this:1.Choose a value for one variable (usuallyx).2.Substitute that value into the equation.3.Solve to find the value ofy.4.Write the solution as anordered pair (x, y).5.Plot that point on the graph.6.Repeat with different values.For example, you might try:(x = 0)(x = 1)(x = 2)Each value gives you another point to plot.After plotting several points, you canconnect them to form the graph of the equation.Example 1:Graphing an EquationLet’s learn how tograph an equation on the coordinate plane.ProblemGraph the equation:[x + y = 6]To graph this equation, we need tofind several pairs of values for x and ythat satisfy the equation.These pairs will becomepoints on the graph.
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Algebra I – Coordinate Geometry - Page 7 preview imageStudy GuideStep 1:Choose a Value for (x)Start by choosing a value forx, then solve the equation to findy.When (x = 0)Substitute into the equation:[0 + y = 6][y = 6]So the coordinate is:[(0, 6)]When (x = 1)Substitute into the equation:[1 + y = 6]Subtract 1 from both sides:[y = 5]So the coordinate is:[(1, 5)]When (x = 2)Substitute into the equation:[2 + y = 6]Subtract 2 from both sides:[y = 4]So the coordinate is:
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Algebra I – Coordinate Geometry - Page 8 preview imageStudy Guide[(2, 4)]Step 2:Organize the Values in a TableMaking a small table can help keep the values organized.Each row in the table gives us apoint to plot on the graph.Step 3:Plot the PointsPlot the following points on the coordinate plane:(0, 6)(1, 5)(2, 4)
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Algebra I – Coordinate Geometry - Page 9 preview imageStudy GuideFigure 4Plotting of coordinates (0,6), (1,5), (2,4)Step 4:Draw the LineAfter plotting the points, draw astraight line that passes through all of them.This line represents thegraph of the equation (x + y = 6).
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Algebra I – Coordinate Geometry - Page 10 preview imageStudy GuideFigure 5The line passing through the plotted pointsWhy Do These Points Form a Line?When the solutions of an equation form astraight line when graphed, the equation is called alinearequation.In this example, the points:(0, 6)(1, 5)(2, 4)all lie on the same straight line.This shows that(x + y = 6) is a linear equation.What Are Nonlinear Equations?
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