Question

Write an equation for a rational function whose graph of the form f(x)=( ax+b)/( cx+d) has all the indicated features . 1) x-intercept 1/4 2) y-intercept 1/2 3) vertical asymptote x=2/3 4) horizontal asymptote y=4/3

Answer 1

To write an equation for a rational function with given features, substitute values into the general form of a rational function and solve the resulting equations.

Step-by-step explanation:

To write an equation for a rational function with the given features, we can substitute the x-intercept, y-intercept, vertical asymptote, and horizontal asymptote into the general form of a rational function: f(x) = (ax + b)/(cx + d).

1) x-intercept: When x = 1/4, the y-value is 0. Substituting these values into the equation, we get 0 = (a * (1/4) + b)/(c * (1/4) + d).

2) y-intercept: When x = 0, the y-value is 1/2. Substituting these values into the equation, we get 1/2 = b/d.

3) Vertical asymptote: When x = 2/3, the function approaches infinity. This means that (2c + d)/(3c + d) = infinity or (2c + d) = infinity.

4) Horizontal asymptote: The horizontal asymptote is y = 4/3, which means that a/c = 4/3.

Solving these equations simultaneously, we can find the values of a, b, c, and d. Once we have the values, we can substitute them into the general equation to get the equation of the rational function.