Question

Determine a possible equation of this rational functions, A rational function which has a vertical asymptote with equation x = 6, a horizontal asymptote with equation y = 0, a y-intercept at the point (0, 0.5) and no x-intercepts.

Answer 1

To determine a possible equation of the given rational function with specific asymptotes and intercepts, consider the given information and construct an equation that satisfies those conditions.

Step-by-step explanation:

To determine a possible equation of the given rational function, we need to consider the given information:

The vertical asymptote is x = 6, which means the function approaches infinity as x approaches 6.

The horizontal asymptote is y = 0, which means the function approaches 0 as x or y approaches infinity.

The y-intercept is (0, 0.5), which means the function crosses the y-axis at y = 0.5 when x = 0.

There are no x-intercepts, which means the function does not cross the x-axis.

Based on this information, a possible equation for the rational function can be:

y = (0.5) / ((x - 6) + 1)

This equation satisfies the given conditions. When x approaches 6, the denominator becomes very small, resulting in a large y value (vertical asymptote). As x or y approaches infinity, the function approaches 0 (horizontal asymptote). When x = 0, the numerator evaluates to 0.5, giving the y-intercept at (0, 0.5). And since the denominator can never be zero, there are no x-intercepts.

Answer 2

To determine a possible equation of the given rational function with specific properties, such as a vertical asymptote, horizontal asymptote, y-intercept, and no x-intercepts, we can use the equation y = a / (x - 6), where 'a' is a constant.

Step-by-step explanation:

To determine a possible equation of the given rational function, we can start by considering the properties of the function. We know that it has a vertical asymptote at x = 6, which means that the function approaches infinity as x approaches 6. We also know that it has a horizontal asymptote at y = 0, which means that the function approaches 0 as x approaches infinity or negative infinity. Additionally, we are given the y-intercept at (0, 0.5) and no x-intercepts.

Based on these properties, a possible equation for the rational function may be:

y = a / (x - 6)

where 'a' is a constant that determines the steepness of the function. The steepness can be adjusted to fit the given properties of the function.

For example, if we choose 'a' to be 3, the equation becomes:

y = 3 / (x - 6)

This equation satisfies the given properties of the function with a vertical asymptote at x = 6, a horizontal asymptote at y = 0, a y-intercept at (0, 0.5), and no x-intercepts.