Final answer:
The equation of the rational function that meets the given conditions of having vertical asymptotes at x = 3 and x = -3, a horizontal asymptote at y = 0, no x-intercepts, and a y-intercept at (0, -1) is y = -1/((x - 3)(x + 3)).
Step-by-step explanation:
To write the equation of the rational function that satisfies the given conditions, we need to consider what those conditions tell us about the function.
First, the function has vertical asymptotes at x = 3 and x = -3. Vertical asymptotes occur where the function is undefined, which often happens when the denominator of a rational function is zero. Therefore, the factors (x - 3) and (x + 3) should appear in the denominator of our function.
Next, the function has a horizontal asymptote at y = 0. This typically means that the degree of the numerator is less than the degree of the denominator in the rational function.
Third, the function has no x-intercepts, indicating that the numerator does not have any real zeros.
Finally, the function has a y-intercept at (0, -1), which means when we plug in x = 0, the function's value should be -1.
Taking all the above into account, a function that meets the criteria is:
y = -1/((x - 3)(x + 3))
This function will have the correct vertical asymptotes and y-intercept, no x-intercepts, and the degree of the numerator (0, since it is a constant) is less than the degree of the denominator (2), giving the horizontal asymptote at y = 0.