2Topics Cartesian Plane Methods of Graphing Intercept Slope Scale First QuadrantInequality GraphsRegion
3Cartesian Plane Named after Rene Deo Cartes a french mathematician Also a philosopher “I think therefore I am”His goal was to create a “picture” that could show a relationship between two variables. We have one for one variable – the number line.
5Some Facts We only need two points to draw a straight line The point where a graph crosses or touches the x axis is called the x interceptIt is found by substituting y = 0The point where a graph crosses or touches the y axis is called the y interceptIt is found by substituting x = 0
6Intercept Method Calculate both intercepts. Place on graph and join Example #1: y = 2x – 6X intercept (y = 0) 0 = 2x – 6-2x = - 6x = 3
7Intercept Method Now find Y intercept Example #1: y = 2x – 6 Y intercept (x = 0) y = 2 (0) – 6y = -6
8Finding X and Y Intercept Example #2: 5x – 3y = 15x int (y = 0) 5x – 3(0) = 155x = 15x = 3y Int (x = 0) 5(0) – 3y = 15-3y = 15y = -5
9Drawing on GraphNow that you know the x & y intercept, you have two points and now can draw the straight line… do it!Practice plotting with other points…
16Understanding the Slope If m or the slope is 2 this means a rise of2 and a run of 1 (2 can be written as 2 )1If m = - 5, this means a rise of -5 and right 1If m= -2 this means rise of -2 right 33
17Understanding the Slope Consider m = -34What is the rise and what is the run?Suggest to put the negative sign on the top to clarify (rise of -3)Numerator always rise (could go up or down)Denominator always run (right only)RiseRun
18Consider y = 2x + 3 What is the slope, rise, run and y intercept? We have a slope 22 can be written as 21Rise of 2Run of 1y intercept of 3 (y = b)Plot on graph paper the following…
19Ex#1: y=2x+3 Question: Draw this line (1,5) 0,3 Where can you plot the y intercept?0,3What is the y intercept?What is the slopeWhat does the slope mean?Up 2, Right 1
20Example #2 y = -5 xWhat is the y intercept, slope? Rise and run?Y intercept is 1Slope is -5/7Rise is – 5Run is 7Plot on graph (put it on graph paper)
21Example #3 y = x What is the y intercept, slope, rise and run? y intercept = 0 (y int let x = 0)Slope = 1Rise of 1Run of 1Plot on graph
22Example #4: 3x – 4y = 12 What is the y intercept, slope, rise and run? Must put in standard form-4y = - 3x + 12y = 3x – 34y intercept = -3Slope ¾Rise of 3, run of 4Plot on graph
23Graphing with ScaleScale is mostly used to make sure your graph can be seenConsider y = 2x + 1003
24Ex#5 y=2x+100 3 How will you measure m = 2/3? Y intercept? (300,300) 500xyHow will you measure m = 2/3?Y intercept?(300,300)Slope?(0,100)You can put 500 along the x axis which means each hash mark is 100Note slope is a ratio so scale does not effect it
25Ex. #5 200x + 300y =300y = - 200xY = -2x + 4003Plot itDo #4 on stencil use form C
261st QuadrantThere will be times when you will need to put the graph only in the 1st quadrantThe problem only exists when the y intercept is negativeIn that case, work with the x intercept (sub y = 0)
27Consider y = 2x – 5 3 Stencil: Do #5 Show how the graph intersects in the 1st quadrantNotice that b is negative.In those cases, work with x int (let y = 0)0 = 2x – 530 = 2x – 15-2x = -15x = 7.5Stencil: Do #5
28Inequality GraphsThe straight line of the graph divides the plane into two regionsOne side will be greater than, one side less than
29The Trick in Standard Form If greater then shade above >If less then shade below <If equal then solid lineIf not equal then dotted line
30Ex y > x + 3 Step 1: Draw Line Y intercept? Slope? 5xyY intercept?Slope?m = 1 (up 1, right 1)Dotted Line or solid?Shade above or below?
31Ex y < x + 3 Step 1: Draw Line Y intercept? Slope? 5xyStep 1: Draw LineY intercept?Slope?m = 1 (up 1, right 1)Dotted Line or solid?Shade above or below?
32Ex 5x - 10y < 30 Step 1: Put in Standard Form 10y < - 5x + 30 2y intercept?Slope?m = 1 (up 1, right 2)Dotted Line or solidline?Do #6 in CShade above or below?
33Point of Intersection If we have two graphs, we create four regions 1 243
34Consider y > 3x – 5 y < -2x + 5 Draw lines… one at a time 2nd line…y intercept of 1st?y int?Slope?Slope?Dotted / solid?m = 3 (3 up, right 1)Above or Below?Dotted Line or solid?Use arrowsShade above or below?Hint… with 2 lines, use arrows at first instead of shadingShade where they intersect!
35Find POI (Point of Intersection), you can also use equations y = 3x – 5y = -2x + 53x – 5 = -2x + 55x = 10x = 2x = 2 y = 3(x) – 5y = 3 (2) – 5y = 1POI (2, 1)Do 7 in EFinish Study Guide