Final answer:
To construct a rational function with the given characteristics, we factor in the locations of vertical asymptotes and x-intercepts into the denominator and numerator respectively, and adjust the constant term to achieve the specified y-intercept, resulting in the function f(x) = (7(x - 6)(x + 6))/((x - 1)(x + 2)).
Step-by-step explanation:
To write an equation for a rational function that meets the given criteria, we need to incorporate vertical asymptotes, x-intercepts, and the y-intercept.
Vertical asymptotes occur where the function is undefined, which means they are values of x for which the denominator of the rational function is zero. Therefore, if we have vertical asymptotes at x = 1 and x = -2, the denominator of our function will be (x - 1)(x + 2).
X-intercepts occur where the function equals zero, which corresponds to the numerator of a rational function being zero. Given x-intercepts at x = 6 and x = -6, the numerator will be (x - 6)(x + 6) or x2 - 36.
Lastly, the y-intercept is the value of the function when x = 0. For our function to have a y-intercept at 7, we need the constant term in the numerator to be 7 times the constant term in the denominator when x = 0. Therefore, we adjust the numerator to get the desired y-intercept.
Putting it all together, our rational function equation is:
f(x) = \frac{7(x - 6)(x + 6)}{(x - 1)(x + 2)}
This function has the correct vertical asymptotes, x-intercepts, and y-intercept at 7.