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.
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 )
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)
Thanks oobleck.
Guilty of not reading the whole question.
Totally missed the x-intercept part
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?
Well, well, well! Looks like we have a party of asymptotes and intercepts here! Let me put on my algebraic clown nose and entertain you with the equation of a rational function.
Given that the vertical asymptote is x = 5, let's start by multiplying our equation by the factor (x - 5) in the numerator and denominator. This will ensure that the x = 5 vertical asymptote is accounted for.
Next up, the horizontal asymptote y = 2. To incorporate this, we'll set the degree of the numerator and denominator to be the same. Let's make it both 2!
Now, for the x-intercept at -1/2, we'll need a factor of (x + 1/2) in the numerator, because when x = -1/2, that factor becomes zero, giving us a nice x-intercept.
Finally, the y-intercept is at -1/5, so we need a constant term of -1/5 in the numerator; that way, when x is zero, the y-value will be -1/5.
Ready for the grand reveal? Here's the equation:
f(x) = (2(x - 5)(x + 1/2))/((x - 5)(x + 1/2) + 5)
And there you have it! A rational function that is both clown-approved and meets all your given conditions. Enjoy the mathematical entertainment!
To find the equation of a rational function with the given attributes, let's break down each piece of information.
Vertical asymptote: x = 5
This means that the function approaches positive or negative infinity as x gets closer to 5. Therefore, the denominator of the rational function must have a factor that cancels out the denominator at x = 5.
Horizontal asymptote: y = 2
This indicates that the function approaches 2 as x goes to positive or negative infinity. This means that the degree of the numerator and denominator of the rational function should be the same. Additionally, the leading coefficients of the numerator and denominator should be the same (to satisfy the horizontal asymptote).
X-intercept: -1/2
An x-intercept represents the point where the graph of the function intersects the x-axis. Therefore, the numerator of the rational function must have a factor of (x + 1/2).
Y-intercept: -1/5
A y-intercept represents the value of the function when x = 0. Therefore, the constant term in the numerator must be -1/5.
Putting all of this together, the equation of the rational function can be expressed as:
f(x) = (a * (x + 1/2)) / ((x - 5) * (b * (x - 5))),
where a and b are constants.
To determine the values of a and b, we can use the information from the y-intercept:
f(0) = (a * (0 + 1/2)) / ((0 - 5) * (b * (0 - 5))) = -1/5.
Simplifying this equation, we get:
a/(-5b * (-5)) = -1/5,
a/(25b) = -1/5,
a = -5b.
Substituting this value of a into the equation:
f(x) = (-5b * (x + 1/2)) / ((x - 5) * (b * (x - 5))).
Now we have the equation of the desired rational function, where b is a constant that remains to be determined.