Question

finding the intercepts asymptotes domain and range from the graph of a rational function

Answer 1

(a) Vertical asymptote: x = 5

Horizontal asymptote: y = 0

(b) Domain: (-∞, 5) ∪ (5, ∞)

Range: (-∞, 0)

(c) x-intercept(s): None

y-intercept: -1

Explanation:

Part (a)

Vertical asymptote

A vertical asymptote is a vertical line that the curve gets infinitely close to, but never touches. It is displayed as a vertical dashed line on the given graph. Therefore, the vertical asymptote is:

• x = 5

Horizontal asymptote

A horizontal asymptote is a horizontal line that the curve gets infinitely close to, but never touches. It is displayed as a horizontal dashed line on the given graph. Therefore, the horizontal asymptote is:

• y = 0

`\hrulefill`

Part (b)

Domain

Since the graph has a vertical asymptote at x = 5, it means that the function is undefined at x = 5. Therefore, the domain of the graph will be all real numbers except x = 5:

• (-∞, 5) ∪ (5, ∞)

Range

Since there is a horizontal asymptote at y = 0 and the curve appears to be always below the x-axis, it indicates that the range of the graph will be all negative y-values. Therefore, the range of the graphed function is:

• (-∞, 0)

`\hrulefill`

Part (c)

x-intercept(s)

The x-intercepts are the x-values of the points where the curve intersects the x-axis, so when the y-coordinate of a point on the graph is zero.

As the given graph has a horizontal asymptote at y = 0 and the curve appears to be always below the x-axis, it implies that the graph does not cross the x-axis. Therefore:

• No x-intercepts

y-intercept(s)

The y-intercept is the y-value at the point where the curve intersects the y-axis, so when the x-coordinate of a point on the graph is zero.

From inspection of the given graph, we can see that the curve crosses the y-axis at y = -1. Therefore:

• y-intercept = -1