Find an equation of a rational function

In summary, to find an equation of a rational function f(x) that satisfies the given conditions, we can use the fraction (x-7)/(x+5)(x) where the x intercept of 7 gives the (x-7) in the numerator and the vertical asymptotes of x = -5 and 0 give the factors of (x+5) and (x-0) in the denominator. The second degree in the denominator vs. first degree in the numerator gives the horizontal asymptote of 0. Finally, multiplying the fraction by a constant a does not change any of the given conditions.
  • #1
DDRchick
27
0
Find an equation of a rational function f(x)that satisfies the conditions. We're allowed to use a calculator, by the way. (:
Okay, conditions:
vertical asymptote: x=-5, x=0
Horizontal asymptote: y=0
x-intercept=7; f(1)=4




On the test i had no idea how to do it, but after seeing her key,I somewhat understand. I looked in the textbook and was able to see where many things came from.
a(x-7)/(x+5)(x)
for one, i have no idea where the a and the x came from. I do, however, understand that the (x-7) is from the x-int. and that the (x+5) is from the v.a.
4=a(1-7)/(1+5)(x) again, no idea where the a and x came from. I see that 4=f(1) which means that y=4 and x=1.
4=a(-6)/(6)(1)
I understand she plugged in that last x finally with a 1 and simplified everything else.
4=-a, a=-4
f(x)=-4(x-7)/x(x+5)
I understand the final equation. I'd just love to know were she got that a and the x (by iself on the denominator) from! Thanks (:
 
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  • #2
The vertical asymptotes of x = -5 and 0 gave the factors of (x+5) and (x-0) in the denominator. The x intercept of 7 gives the (x-7) in the numerator.

That's where the fraction [itex]\frac{x-7}{(x+5)(x)}[/itex] comes from. The second degree in the denominator vs. first degree in the numberator gives the horizontal asymptote of 0 for free. You have one condition left and only need to note that multiplying the fraction by a constant a doesn't change any of the above features and let's you get the last constraint.
 
  • #3
Ohhh okay thanks so much! :]
 

Related to Find an equation of a rational function

1. What is a rational function?

A rational function is a mathematical expression that can be written as a ratio of two polynomials. It typically takes the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

2. How do I find the equation of a rational function?

To find the equation of a rational function, you need to determine the values of the numerator and denominator polynomials. This can be done by plugging in a set of x-values and solving for the corresponding y-values. Once you have a set of points, you can use them to create a table and then find the equation using a graphing calculator or by hand.

3. What is the process for graphing a rational function?

To graph a rational function, you first need to find the x-intercepts and asymptotes of the function. The x-intercepts are the points where the function crosses the x-axis, and the asymptotes are the lines that the function approaches but never touches. Once you have these points, you can plot them on a graph and draw a curve connecting them.

4. How do I know if a point is on the graph of a rational function?

To determine if a point is on the graph of a rational function, you can plug in the x-value of the point into the function and see if the resulting y-value is equal to the y-value of the point. If they are equal, then the point is on the graph.

5. Can a rational function have more than one horizontal asymptote?

Yes, a rational function can have more than one horizontal asymptote. This can occur when the degree of the numerator is equal to or greater than the degree of the denominator. In this case, the horizontal asymptotes can be found by dividing the leading coefficients of the numerator and denominator.

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