Question

Write an equation for a rational function with:

Vertical asymptotes at x=3 and x=−5
x intercepts at x=−2 and x=5
y intercept at 6
y=__

Answer 1

To create an equation for a rational function with specific features, you can use the factored form of the equation and substitute the given values. In this case, the equation is f(x) = -3.75(x + 2)(x - 5) / ((x - 3)(x + 5)).

Step-by-step explanation:

To create an equation for a rational function with vertical asymptotes at x = 3 and x = -5, x intercepts at x = -2 and x = 5, and a y-intercept at y = 6, we can start by writing the equation in factored form:

f(x) = a(x - x1)(x - x2) / ((x - x3)(x - x4))

Where x1, x2, x3, and x4 are the x-values of the vertical asymptotes and the x-intercepts, and a is a constant.

In this case, x1 = -2, x2 = 5, x3 = 3, and x4 = -5. Substituting these values into the equation gives:

f(x) = a(x + 2)(x - 5) / ((x - 3)(x + 5))

Finally, to find the value of a, we can substitute the y-intercept (0, 6) into the equation:

6 = a(0 + 2)(0 - 5) / ((0 - 3)(0 + 5))

Solving for a, we get a = -3.75

Therefore, the equation for the rational function is:

f(x) = -3.75(x + 2)(x - 5) / ((x - 3)(x + 5))