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.
5 answers
could be as simple as y = 2x/(x-5) - 1/5
not quite. The above has an x-intercept of x=0
y = (2x+1)/(x-5)
y = (2x+1)/(x-5)
Thanks oobleck.
Guilty of not reading the whole question.
Totally missed the x-intercept part
Guilty of not reading the whole question.
Totally missed the x-intercept part
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 )
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 )
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?
What could you do to change the y-intercept to, say, y = -1?