let f(x)=(x^2+2x-15)/(3x^2+12x-15)
Find the horizontal and vertical asymptotes of f(x).
To have a vertical asymptote , the denominator must be zero
so, 3x^2 + 12x - 15 = 0
x^2 + 4x - 5 = 0
(x+5)(x-1) = 0
x = -5 or x = 1
so we have two vertical asymptotes,
x = -5 and x = 1
for the horizontal asymptote, let x become large
then we are left with x^2/3x^2 or 1/3
V.A. : y = 1/3
I did the exact same thing as you did for the verticle asymptotes, but my online homework says that -5, or 1 are wrong. I even tried putting both of them in list form.
sorry, but I am 100% confident that what I did was correct. Since I can't see what type of input your online website expects, we are at an impasse.
To find the horizontal and vertical asymptotes of a function, we need to analyze the behavior of the function as x approaches infinity or negative infinity.
1. Horizontal asymptote:
- A horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator.
- If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
- For our function f(x):
- The degree of the numerator is 2 (highest power of x is x^2).
- The degree of the denominator is also 2 (highest power of x is 3x^2).
- Therefore, the degrees are equal.
- To find the horizontal asymptote, divide the leading coefficients of the numerator and denominator.
- The leading coefficient of the numerator is 1 (the coefficient of x^2).
- The leading coefficient of the denominator is 3 (the coefficient of 3x^2).
- The ratio of the leading coefficients is 1/3.
Therefore, the horizontal asymptote of f(x) is y = 1/3.
2. Vertical asymptote:
- A vertical asymptote occurs when the denominator becomes zero.
- To find the vertical asymptotes, set the denominator equal to zero and solve for x.
- In our case, the denominator is 3x^2 + 12x - 15.
3x^2 + 12x - 15 = 0
- Now, we solve this quadratic equation either by factorization or by using the quadratic formula:
- Factorization: This equation can be factored as (3x - 3)(x + 5) = 0.
- Setting each factor equal to zero gives us two possible solutions: x = 1 and x = -5.
- Therefore, there are two vertical asymptotes: x = 1 and x = -5.
In summary, the horizontal asymptote of f(x) is y = 1/3, and the vertical asymptotes are x = 1 and x = -5.