Question

Graph the function. Describe the domain. $$y=\frac{-5 x+19}{x-3}$$

Short answer

The domain of the function is (-∞, 3) union (3, ∞). And the function has a vertical asymptote at x = 3 and a horizontal asymptote at y = -5.

Step-by-step solution

Identify the Domain

For any rational function in the form \(y= \frac{ax+b}{x-c}\), a point is not in the domain if it makes the denominator zero. Here, x-3 in the denominator will be zero when x = 3. Hence, the domain of the function will be all real numbers except x = 3. Or we can say the domain is (-∞, 3) union (3, ∞).

Identify Vertical Asymptote

A vertical asymptote of a function is a vertical line x=d where the function approaches +∞ or -∞ as x approaches d. Here, we have a vertical asymptote at x-3=0, or x=3, the point that is not part of the domain.

Identify Horizontal Asymptote

A horizontal asymptote of a function is a horizontal line y=e where the function approaches e as x approaches +∞ or -∞. By looking at the degrees of the polynomial in the numerator and the denominator, we can tell that if the degrees are the same, the horizontal asymptote is y=a, where a is the ratio of the leading coefficients. Here, both the numerator and the denominator are first degree polynomials, so there will be a horizontal asymptote at y=-5.

Graph the Function

With the domain, vertical and horizontal asymptotes known, we can then draw the graph. It should be noted that the graph approaches but never touches or crosses the asymptotes. So it is a hyperbola that opens downwards and sideways and is centered at x = 3. It approaches the horizontal asymptote , but never actually meets it.