Find an equation for a rational function with the following properties:
vertical asymptotes at x=1 and x=−9
zero at x=11
f(0)=15
horizontal asymptote at y=0
please check my work thanks
I got:
15(x-11)^2/(x-1).(x+9)
too bad you didn't go to an online graphing site to check it out
15(x-11)^2/(x-1).(x+9)
has a horizontal asymptote of y=15 since top and bottom have the same degree.
If you want the asymptote at y=0, you need the top to have a lower degree than the bottom. So, I'd start with
y = (x-11)/(x-1).(x+9)
Now, that has f(0) = -11/(-1*9) = 11/9
So you need 15 * 9/11 (x-11)/(x-1).(x+9) = 135(x-11) / 11(x-1)(x+9)
the graph is at
www.wolframalpha.com/input/?i=135%28x-11%29+%2F+%2811%28x-1%29%28x%2B9%29%29+for+-10%3C%3Dx%3C%3D10
Thank you, oobleck
To find an equation for a rational function with the given properties, you need to consider the characteristics of a rational function that can result in the desired asymptotes and zero.
First, let's start with the vertical asymptotes. We know that the function has vertical asymptotes at x=1 and x=-9. This means that the function should have factors of (x-1) and (x+9) in the denominator.
Next, we know that the function has a zero at x=11. This means that the numerator should have a factor of (x-11), as this will make the function equal to zero when x=11.
Now, we have a function in the form: f(x) = (x-11) / [(x-1)(x+9)]
However, we still need to consider the horizontal asymptote at y=0. For a rational function, the degree of the numerator should be less than or equal to the degree of the denominator for the function to have a horizontal asymptote at y=0. In our current equation, the degree of the numerator (1) is less than the degree of the denominator (2). Therefore, we need to modify the function by multiplying by a constant to make the numerator degree equal to the denominator degree.
To achieve this, we can multiply the numerator by some constant k. Let's denote k as k(x-11) and rewrite the function:
f(x) = k(x-11) / [(x-1)(x+9)]
Now, let's substitute f(0) = 15 into our equation. We get:
15 = k(0-11) / [(0-1)(0+9)]
15 = -11k / (-1)(9)
15 = -11k / -9
15 = (11/9)k
To solve for k, we can multiply both sides of the equation by 9/11:
15 * 9/11 = k
135/11 = k
Substituting this value of k back into the equation, we finally get the equation of the rational function:
f(x) = (135/11)(x-11) / [(x-1)(x+9)]
So, the correct equation for the rational function with the given properties is:
f(x) = (135/11)(x-11) / [(x-1)(x+9)]