Question

State the equation of a rational function if the vertical asymptote is x = 5, the horizontal asymptote is y = 2, the x-intercept is -1/2 and the y-intercept is -1/5.

Answer 1

could be as simple as y = 2x/(x-5) - 1/5

Answer 2

not quite. The above has an x-intercept of x=0

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

Answer 3

Thanks oobleck.
Guilty of not reading the whole question.
Totally missed the x-intercept part

Answer 4

Detailed explanation.

Rational function can be written in the form:

f(x) = P(x) / Q(x)

Vertical Asymptote is point where denominator is equal zero.

Vertical asymptote at x = 5

This requires a factor of ( x - 5 ) in the denominator because:

For x = 5

x - 5 = 0

Horizontlal Asymptote is point where f(x) = 0

This requires a factor of 2 x in the numerator because:

For x = 2 , f(x) = 0

2 / ( x - 5 )

The x-intercept is the point at which the graph crosses the x-axis.

At this point, the f(x) is zero.

The x-intercept is - 1 / 2

This requires a factor of ( x + 1 / 2 ) in the numerator because:

For x = - 1 / 2 , ( x + 1 / 2 ) = 0

2 ( x + 1 / 2 ) / ( x - 5 )

( 2 x + 1 ) / ( x - 5 )

The degrees of the numerator and denominator are the same at this point.

You just need to add constant factors to make the ratio of the leading coefficients equal to 2 / 1.

The ratio of leading coefficients is already 2 / 1

So:

f(x) = ( 2 x + 1 ) / ( x - 5 )

Answer 5

Extra credit: Luckily, this simple function has a y-intercept of -1/5 as required.
What could you do to change the y-intercept to, say, y = -1?