Unit D: Graphing

Summary


Predicting the Shape of a Graph in Real-Life Situations

  • Trends of linear relations can show positive correlation or negative correlation.
  • A graph that represents direct variation and positive correlation can look like the following:







  • A graph that represents partial variation can look like the following:







  • A graph that shows negative correlation can look like the following:







The Slope of a Line

  • The slope is the steepness of a line on a graph.
  • On a graph, the slope is calculated using the following formula:
    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»m«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»rise«/mi»«mi»run«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»change«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»y«/mi»«/mrow»«mrow»«mi»change«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
  • When two points are known, the slope is calculated using
    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨» «mi»m«/mi» «mo»=«/mo» «mfrac» «mrow» «msub» «mi»y«/mi» «mn»2«/mn» «/msub» «mo»-«/mo» «msub» «mi»y«/mi» «mn»1«/mn» «/msub» «/mrow» «mrow» «msub» «mi»x«/mi» «mn»2«/mn» «/msub» «mo»-«/mo» «msub» «mi»x«/mi» «mn»1«/mn» «/msub» «/mrow» «/mfrac» «/math»
  • When a linear relation demonstrates direction variation,
    • the graph is a straight line
    • the graph contains the ordered pair (0, 0), which is the origin
    • the slope, or rate of change, is constant
  • When a linear relation demonstrates partial variation,
    • the graph is a straight line
    • the graph does not include the ordered pair (0, 0)
    • the slope, or rate of change, is constant

Writing the Equation for a Linear Relationship

  • In any direct variation relationship, the equation of the line is
    y = mx, where m is the slope, or rate of change
  • In any partial variation relationship, the equation of the line is
    y = mx + b, where m is the slope, or rate of change, and b is the y-intercept

Using Graphs to Estimate Values

  • Interpolation uses known data points to interpret the relationship in between data points. A new data point can be estimated within the range of data collected when using the method of interpolation.
  • Extrapolation uses data and graphs to predict new data beyond the range of data collected. The line is extended to provide an estimated value beyond the given points when using the method of extrapolation.