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 [4 MARKS - TI]

Answer 1

The equation for the rational function with the given features is f(x) = (4x+3/2)/(3x+3), derived from setting the x-intercept at 1/4, the y-intercept at 1/2, the vertical asymptote at x = 2/3, and the horizontal asymptote at y = 4/3.

Step-by-step explanation:

To find an equation for a rational function that has an x-intercept of 1/4, a y-intercept of 1/2, a vertical asymptote at x=2/3, and a horizontal asymptote at y=4/3, we can set up the general form f(x) = (ax+b)/(cx+d). The x-intercept at x=1/4 implies that when f(x) = 0, x = 1/4, so the numerator must be zero when x = 1/4, which suggests that a=b. The y-intercept at (0, 1/2) implies that f(0) = 1/2, allowing us to set up the equation b/d = 1/2. The vertical asymptote at x = 2/3 indicates that the denominator of f(x) should be zero when x = 2/3, hence c=3d/2. The horizontal asymptote at y = 4/3 suggests the leading coefficients of the numerator and denominator must have the ratio of 4:3, leading us to a=4d/3.

Now we have a system of equations: a=b, b/d=1/2, and a=4d/3. Solving this system, we can choose d=3, b=3/2, a=4, and c=3. Thus, the function is f(x) = (4x+3/2)/(3x+3).