algebra 2 notes (graphing)

  • Quadrants and Curves

    • First and third quadrants contain two curves.
    • Curves drawn in second and fourth quadrants.

  • Asymptotes

    • Definition: Boundary lines for curves.
    • Vertical asymptote: Drawn as a dotted vertical line.
    • Horizontal asymptote: Drawn as a dotted horizontal line.

  • Curve Placement

    • Positive numerator: Curves in first and third quadrants.
    • Negative numerator: Curves in second and fourth quadrants.

  • Finding Asymptotes

    • Vertical asymptote (h) = opposite of the value in the denominator.
    • Example: From x−5x - 5, take h=5h = 5.
    • Horizontal asymptote (k) is just the constant outside the fraction.
    • Example: y=6y = 6 if constant is positive six.

  • Domain and Range

    • Domain: All numbers except for the vertical asymptote (h).
    • Range: All numbers except for the horizontal asymptote (k).
    • Example: If h=3h = 3, Domain: all xx except 33.
    • Example: If k=1k = 1, Range: all yy except 11.

  • Graphing Steps

    • Draw coordinate plane with asymptotes.
    • Vertical asymptote: Locate hh on the x-axis and draw dashed line.
    • Horizontal asymptote: Locate kk on the y-axis and draw dashed line.
    • Determine curve placement based on the sign of the numerator.
    • Curves should approach but never cross the asymptotes.

  • Practice Examples

    • Use the process outlined to graph and identify as shown in examples 1-4.
    • Focus on signs and opposites when determining asymptote values.