what is the rational equation of a function with intercepts at (8,0) and (0,0.16) and a horizontal asymptote at y=0.5 and a vertical asymptote at x=3 and a removable discontinuity at (3,-1)
hits y axis at x = 8
hits x axis at Y = .16
y = m x + b
when x = 0, y = .16
so
y = m x + .16
now when y = 0 , x = 8
so
0 = 8 m + .16
so m = - .16/8 = -.02
so
y = -.02 x + .16
or
100 y = -2 x + 16
You cannot have a vertical asymptote at x=3 and a removable discontinuity at x=3.
To find the rational equation of a function with the given properties, we can start by considering the intercepts and asymptotes.
1. Intercepts:
Since the function has intercepts at (8,0) and (0,0.16), we know that it crosses the x-axis at x = 8 and the y-axis at y = 0.16. This gives us two terms in the rational equation: (x - 8) and (y - 0.16).
2. Horizontal Asymptote:
The function has a horizontal asymptote at y = 0.5. For a rational function, the degree of the numerator is either equal to or less than the degree of the denominator. Since the denominator degree determines the behavior of the rational function as x approaches infinity or negative infinity, we can conclude that the degree of the denominator must be at least 1. Therefore, the numerator must have a degree of 0 (no x term). So, the rational equation has a constant numerator.
3. Vertical Asymptote:
The function has a vertical asymptote at x = 3. This means that the denominator of the rational equation must have a factor of (x - 3), as this will cause the expression to approach infinity as x approaches 3.
4. Removable Discontinuity:
The function has a removable discontinuity at (3,-1). This means that there is a factor of (x - 3) both in the numerator and denominator, which cancels out, allowing the function to be defined at x = 3.
Putting all this together, the rational equation can be written as:
f(x) = [(x - 8) * (y - 0.16)] / [(x - 3) * (x - 3)]
Simplifying, we get:
f(x) = (xy - 0.16x - 8y + 1.28) / (x^2 - 6x + 9)
So, the rational equation of the function with the given properties is f(x) = (xy - 0.16x - 8y + 1.28) / (x^2 - 6x + 9).