Determine an equation of a rational function of the form with x-intercept of 1/4, y-intercept of -1/2, vertical asymptote at x = -2/3, and
horizontal asymptote at y = 4/3.
vertical asymptote at x = -2/3 ... 1/(3x+2)
x-intercept of 1/4 ... (4x-1)/(3x+2)
horizontal asymptote at y = 4/3 ... the same
now, y(0) = -1/2 so that condition is also met.
verify at
https://www.wolframalpha.com/input/?i=y%3D%284x-1%29%2F%283x%2B2%29
To determine an equation of a rational function with specific properties, we can follow these steps:
Step 1: Determine the Vertical Asymptote
Given that the vertical asymptote is at x = -2/3, we can set the denominator of the rational function equal to zero and solve for x:
x + 2/3 = 0
x = -2/3
Step 2: Determine the Horizontal Asymptote
Given that the horizontal asymptote is at y = 4/3, we can determine the end behavior of the rational function. Since the degree of the numerator and denominator is the same (both are linear in this case), the horizontal asymptote is given by the ratio of the leading coefficients. Therefore, the equation for the horizontal asymptote is y = 1/1 = 1.
Step 3: Determine the x-intercept
Given that the x-intercept is at 1/4, we can set the numerator of the rational function equal to zero and solve for x:
x - 1/4 = 0
x = 1/4
Step 4: Determine the y-intercept
Given that the y-intercept of the rational function is at -1/2, we can substitute x = 0 into the equation:
y = (0 - 1/4) / (0 + 2/3) = -1/4 / 2/3
y = -1/4 * 3/2 = -3/8
Step 5: Write the Equation
Now that we have the necessary information, we can write the equation of the rational function in the form of y = (Ax + B) / (Cx + D):
Vertical asymptote: x = -2/3
Horizontal asymptote: y = 4/3
x-intercept: x = 1/4
y-intercept: y = -1/2
The equation of the rational function satisfying the given conditions is:
y = (3x - 3/4) / (2x + 3/2)