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

### 3 Answers

- atsuoLv 63 months agoFavorite 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).

- billrussell42Lv 73 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

- fcas80Lv 73 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.