Math 374 Graphs

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

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.

Notes We recall - 2 -1 0 1 2 3

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

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

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

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

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…

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

Standard Form Method Slope Y intercept 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

Identifying Slant and Slope Y Int Slant Slant Y Int

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

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

Rise, Run & Slope Slope Slope Rise Rise Run Run

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

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

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…

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

Example #2 y = -5 x + 1 7 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)

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

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

Graphing with Scale Scale is mostly used to make sure your graph can be seen Consider y = 2x + 100 3

Ex#5 y=2x+100 3 How will you measure m = 2/3? Y intercept? (300,300) 500 x y How 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 100 Note slope is a ratio so scale does not effect it

Ex. #5 200x + 300y = 120000 300y = - 200x + 120000 Y = -2x + 400 3 Plot it Do #4 on stencil use form C

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

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

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

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

Ex y > x + 3 Step 1: Draw Line Y intercept? Slope? 5 x y Y intercept? Slope? m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

Ex y < x + 3 Step 1: Draw Line Y intercept? Slope? 5 x y Step 1: Draw Line Y intercept? Slope? m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

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

Point of Intersection If we have two graphs, we create four regions 1 2 4 3

Consider 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 arrows Shade above or below? Hint… with 2 lines, use arrows at first instead of shading Shade where they intersect!

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