Final answer:
The equation of the rational function with vertical asymptotes at x = -2 and x = 3, x-intercepts at (-1,0) and (5,0), and y-intercept at (0,5) is y = (-5)(x + 1)(x - 5) / ((x + 2)(x - 3)).
Step-by-step explanation:
To write the equation of a rational function with the given characteristics, we need to consider the requirements for vertical asymptotes, x-intercepts, and the y-intercept. Vertical asymptotes occur where the function is undefined, typically where the denominator of a rational function equals zero. Hence, for vertical asymptotes at x = -2 and x = 3, the factors (x + 2) and (x - 3) must be in the denominator.
X-intercepts are points where the function crosses the x-axis, which means the y-value is 0. This occurs when the numerator of the function equals zero. Given the x-intercepts at (-1,0) and (5,0), (x + 1) and (x - 5) must be factors in the numerator.
The y-intercept is given by the point (0,5), which means when x = 0, y must equal 5. To ensure the correct y-intercept, we adjust the constant factor in the numerator. To get a y-intercept of 5, we multiply the numerator by 5 since the denominator equals -2 * 3 = -6 at x = 0, and 5 / -6 simplifies to an unwanted negative value. Hence, we multiply by -5 to get the positive y-intercept.
Putting this all together, the equation of the rational function is:
y = (-5)(x + 1)(x - 5) / ((x + 2)(x - 3)).