1FAddingHyperbolas

Graphing the Sum of Hyperbolas

math y = \dfrac {a}{f(x)} + \dfrac {b}{g(x)} math

Graphs in this form will have vertical asymptotes where f(x )= 0 and where g(x) = 0 and a horizontal asymptote at y = 0.

BUT the graph will __cross__ the horizontal asymptote at the point where: math \dfrac {a}{f(x)} + \dfrac {b}{g(x)} = 0 math

Example 1

math y = \dfrac {1}{x+2} + \dfrac {1}{x-3} math

This will have asymptotes at x = –2, x = 3, y = 0

The graph will cross the x-axis (asymptote) when: math \dfrac {1}{x+2} + \dfrac {1}{x-3} = 0 math math \dfrac {1}{x+2} = \dfrac {-1}{x-3} math math \\ x - 3 = -1(x + 2) \\ \\ x - 3 = -x - 2 \\ \\ 2x = 1 \\ \\ x = \dfrac {1}{2} math

To sketch math y = \dfrac {1}{x+2} + \dfrac {1}{x-3} math

First sketch the two individual hyperbolas Then sketch the final graph by addition of ordinates Asymptotes x = –2, x = 3, y = 0

math \text {Domain: } x \in R \backslash \lbrace -2, 3 \rbrace math

math \text {Range: } \: y \in R math

x-intercept math x = \frac {1}{2} math

y-intercept math y = \frac {1}{6} math

See Sketching Graphs using Partial Fractions

Return to Summary of Coordinate Geometry

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