How do you graph a function of the form (ax+b)/(cx+d)?

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  • atsuo
    Lv 6
    3 months ago
    Favorite Answer

    The graph y = (ax+b)/(cx+d) : Some cases exist.

    Case1. c = 0

    Case1-1. d ≠ 0

    y = (ax+b)/d = (a/d)x + b/d

    The graph becomes a line, its slope is a/d and its y-intercept is b/d.

    Case1-2. d = 0

    The denominator cx+d always becomes 0, so no graph exists.

    Case2. c ≠ 0

    y = ((a/c)x + b/c)/(x + d/c)

    = ((a/c)(x + d/c) - (a/c)(d/c) + b/c)/(x + d/c)

    = a/c + (b/c - ad/c^2)/(x + d/c)

    = a/c + K/(x + d/c) (K = b/c - ad/c^2 is a constant.)

    Case2-1. K ≠ 0 (b ≠ ad/c)

    Write a graph y = K/x and shift it left by d/c and shift it up by a/c. That is, the vertical asymptote is x = -d/c and the horizontal asymptote is y = a/c.

    Case2-2. K = 0 (b = ad/c)

    The graph becomes a horizontal line y = a/c,  but it has a hole at x = -d/c because K/(x + d/c) becomes 0/0 (undetermined).

  • 3 months ago

    you need values for the constants, and assuming it is y = (ax+b)/(cx+d)

    then either do it point by point, or use a graphing web site.

    y = (x+2)/(3x+4) is below

    Attachment image
  • fcas80
    Lv 7
    3 months ago

    It will be two branches, sort of like a hyperbola.  It will be discontinuous for x where the denominator becomes zero, and then y becomes large positive or large negative depending on the branch.  When x gets very large positive or large negative, y will approach a/c.

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