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Math 374 Graphs

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Topics Cartesian Plane Methods of Graphing Intercept Slope Scale First Quadrant Inequality Graphs Region

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Cartesian 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.

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Notes We recall

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Some 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 intercept It is found by substituting y = 0 The point where a graph crosses or touches the y axis is called the y intercept It is found by substituting x = 0

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Intercept Method Calculate both intercepts. Place on graph and join Example #1: y = 2x – 6 X intercept (y = 0) 0 = 2x – 6 -2x = - 6 x = 3

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Intercept Method Now find Y intercept Example #1: y = 2x – 6 Y intercept (x = 0) y = 2 (0) – 6 y = -6

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Finding X and Y Intercept Example #2: 5x – 3y = 15 x int (y = 0) 5x – 3(0) = 15 5x = 15 x = 3 y Int (x = 0) 5(0) – 3y = 15 -3y = 15 y = -5

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Drawing on Graph Now 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…

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Plotting. C (-4, -4).... B (-4, 2) D (2, -2) (A 3, 1) Q1 (+,+) Q4 (+,-) Q3 (-,-) Q2 (-,+) A (3,1) B (-4,2) C (-4, -4) D (2, -2)

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Standard Form Method All straight lines have a y intercept and a slant called a slope. If the relationship is in standard form we can write it… y = m x + b Slope Y intercept

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Y Int Slant Y Int Identifying Slant and Slope

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Standard Form Recall y = mx + b Dependent Variable (DV) Slope Independent Variable (IV) Y Intercept or Starting Value

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Relationship of y & b It is easy to see how b is the y intercept; we substitute x = 0 x = 0 y = m(0) + b y = b

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Rise, Run & Slope Slope Rise Run Rise Run Slope

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Understanding the Slope If m or the slope is 2 this means a rise of 2 and a run of 1 (2 can be written as 2 ) 1 If m = - 5, this means a rise of -5 and right 1 If m= -2 this means rise of -2 right 3 3

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Understanding the Slope Consider m = -3 4 What 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) Rise Run

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Consider y = 2x + 3 What is the slope, rise, run and y intercept? We have a slope 2 2 can be written as 2 1 Rise of 2 Run of 1 y intercept of 3 (y = b) Plot on graph paper the following…

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Ex#1: y=2x+3 0,3 (1,5) Question: Draw this line What is the y intercept? What is the slope What does the slope mean? Where can you plot the y intercept? Up 2, Right 1

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Example #2 y = -5 x What is the y intercept, slope? Rise and run? Y intercept is 1 Slope is -5/7 Rise is – 5 Run is 7 Plot on graph (put it on graph paper)

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Example #3 y = x What is the y intercept, slope, rise and run? y intercept = 0 (y int let x = 0) Slope = 1 Rise of 1 Run of 1 Plot on graph

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Example #4: 3x – 4y = 12 What is the y intercept, slope, rise and run? Must put in standard form -4y = - 3x + 12 y = 3x – 3 4 y intercept = -3 Slope ¾ Rise of 3, run of 4 Plot on graph

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Graphing with Scale Scale is mostly used to make sure your graph can be seen Consider y = 2x

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Ex#5 y=2x Ex#5 y=2x (0,100) (300,300) 500 x y You can put 500 along the x axis which means each hash mark is 100 Y intercept? Slope? How will you measure m = 2/3? Note slope is a ratio so scale does not effect it

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Ex. #5 200x + 300y = y = - 200x Y = -2x Plot it Do #4 on stencil use form C

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1 st Quadrant There will be times when you will need to put the graph only in the 1 st quadrant The problem only exists when the y intercept is negative In that case, work with the x intercept (sub y = 0)

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Consider y = 2x – 5 3 Show how the graph intersects in the 1 st quadrant Notice that b is negative. In those cases, work with x int (let y = 0) 0 = 2x – = 2x – 15 -2x = -15 x = 7.5 Stencil: Do #5

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Inequality Graphs The straight line of the graph divides the plane into two regions One side will be greater than, one side less than

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The Trick in Standard Form If greater then shade above > If less then shade below < If equal then solid line If not equal then dotted line

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Ex y > x x y Y intercept? Slope? Step 1: Draw Line m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

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Ex y < x x y Y intercept? Slope? Step 1: Draw Line m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

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Ex 5x - 10y < 30 5 x y y intercept? Slope? Step 1: Put in Standard Form m = 1 (up 1, right 2) Dotted Line or solidline? Shade above or below? -10y < - 5x + 30 y > 1x – 3 2 Do #6 in C

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Point of Intersection If we have two graphs, we create four regions

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Consider y > 3x – 5 y < -2x + 5 Consider y > 3x – 5 y < -2x x y Draw lines… one at a time Slope? Hint… with 2 lines, use arrows at first instead of shading m = 3 (3 up, right 1) Dotted Line or solid? Shade above or below? y intercept of 1 st ? 2 nd line… y int? Slope? Dotted / solid? Use arrows Above or Below? Shade where they intersect!

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Find POI (Point of Intersection), you can also use equations y = 3x – 5 y = -2x + 5 3x – 5 = -2x + 5 5x = 10 x = 2 x = 2 y = 3(x) – 5 y = 3 (2) – 5 y = 1 POI (2, 1) Do 7 in E Finish Study Guide

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