Calculators/Mathematics/Homographic function

Homographic function

asymptotes · center of symmetry · domain · value at a point

y =

x +

Vertical asymptote

x = 1

Horizontal asymptote

y = 1

Center of symmetry: (1, 1) · Domain: x ≠ 1 · Range: y ≠ 1

Value at x =

y = 2.5

The graph of a homographic function is a hyperbola shifted by the vector [−d/c, a/c]. Condition for being homographic: c ≠ 0 and ad − bc ≠ 0 (here ad − bc = -3).

A homographic (rational) function has the form:

$y=\frac{ax+b}{cx+d},\quad c\neq 0,\ ad-bc\neq 0$

Its graph is a hyperbola obtained by shifting $y=\tfrac{k}{x}$ by the vector $\left[-\tfrac{d}{c},\ \tfrac{a}{c}\right]$ . It has two asymptotes:

vertical: $x=-\dfrac{d}{c}$ (the zero of the denominator),
horizontal: $y=\dfrac{a}{c}$ .

Their intersection is the center of symmetry of the hyperbola. The domain is $x\neq-\tfrac{d}{c}$ , and the range is $y\neq\tfrac{a}{c}$ . The condition $ad-bc\neq 0$ guarantees that the function does not reduce to a constant.

Examples

y = (x+2)/(x−1)

→ asymptotes: x=1, y=1; center (1,1)

Value at x=3

→ y = (3+2)/(3−1) = 2.5

Frequently asked questions

How do you find the asymptotes of a homographic function?+

The vertical asymptote is x = −d/c (where the denominator = 0), and the horizontal one is y = a/c. Their intersection is the center of symmetry of the hyperbola, i.e. the shift vector relative to y=k/x.

When is a function homographic?+

When c ≠ 0 (otherwise it is linear) and ad − bc ≠ 0 (otherwise it reduces to a constant). The graph is then a hyperbola with two asymptotes.

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