Rational function equation help

Question

A function of the form y=a/bx+c has a y-intercept of -2/5, a vertical asymptote of x=2.5

determine the equation of this function.

Answer 1

y intercept = -2/5

means when x= 0, y= -2/5. It's the point (0, -2/5)

y = -2/5 = 1/(x-2.5) = 1/(0-2.5) = 1/-2.5 = -2/5

y= 1/(x-2.5) has a vertical asymptote x=2.5 as that makes the denominator =0

y intercept = -2/5 when x = 0

y= 1/(0-2.5) = 1/-2.5 = -1/2.5 = -2/5

the equation with y intercept =-2/5 and asymptote x=2.5

is

y= 1/(x-2.5)

or y= 1/(x-5/2) = 1/(2x/2 -5/2) = 1/((2x-5)/2) = 2/(2x-5)

y= 2/(2x-5)

or if you don't like fractions

2xy -5y = 2

when x= 0, -5y=2 and y = -2/5 = the y intercept

when x = 2.5 = 5/2, 2(5/2)y -5y = 2

5y-5y = 2

0 = 2

which is a contradiction

x=2.5= 5/2 is an asymptote

Answer 2

First, you know there exists a vertical asymptote when plugging the x-value into the denominator yields a zero in the denominator. Thus, to ensure that 2.5 is a VA, we must factor out the denominator:

y = a / (b(x+c/b)) This form tells us that c/b must equal -2.5, since plugging in 2.5 for x would cause the denominator to equal zero.

Next, we know that when y = -2.5, x= 0 since the y-intercept is -2.5. Thus, plugging in the point (0,-2.5):

-2.5 = a / (b(0-2.5) which can be simplified to a/b = 6.25

Now since y = a / (b(x+c/b)) which can be written as ( a / b) * (1 / (x + c/b)), we can fill in the known values of a/b and c/b to yield y = 6.25 / (x-2.5)