How do you find the vertical and horizontal assymptotes of the graph of the rational function?
f(x)= 3x^2-5x+2/6x^2-5x+1
f(x)= 2x+3/(�ãx^2-2x+3)
Vertical asymptotes in rational functions usually come from the denominator approaching zero.
In
f(x)= (3x^2-5x+2)/(6x^2-5x+1)
the denominator can be factorized as:
(3x-1)(2x-1)
at x=1/3 or x=1/2, the denominator becomes zero, and hence a vertical asymptote. You also need to check if the numerator vanishes at the same points, in which case the point will be undefined.
To find the horizontal asymptotes, do a long division, and the resulting leading constant (1/2) is the horizontal asymptote. Take care to determine from which side of y=1/2 the curve approaches the asymptote as x approaches ±∞.
To find the vertical and horizontal asymptotes of a rational function, you need to examine the behavior of the function as x approaches positive or negative infinity.
1. Vertical Asymptotes:
To find the vertical asymptotes, set the denominator equal to zero and solve for x. The vertical asymptotes occur at these x-values where the function is undefined. In other words, the vertical asymptotes represent the values that x cannot approach.
For the first function, f(x) = (3x^2-5x+2)/(6x^2-5x+1), we need to solve the quadratic equation 6x^2-5x+1 = 0 to find the vertical asymptotes.
Solving the equation, we get x = 1/2 as the only solution. So, there is a vertical asymptote at x = 1/2 for the first rational function.
For the second function, f(x) = (2x+3) / (x^2 - 2x + 3), we need to solve the quadratic equation x^2 - 2x + 3 = 0 to find the vertical asymptotes.
Since this quadratic equation has no real solutions (the discriminant is negative), there are no vertical asymptotes for the second rational function.
2. Horizontal Asymptotes:
To find horizontal asymptotes, we look at the degrees of the polynomials in the numerator and denominator. The degrees determine the behavior of the function as x approaches positive or negative infinity.
For the first function, both the numerator and denominator are quadratic polynomials with the same leading coefficient. In this case, when the degrees are equal, the horizontal asymptote occurs at the ratio of the leading coefficients. So, the horizontal asymptote is y = 3/6, which simplifies to y = 1/2.
For the second function, the numerator is a linear polynomial and the denominator is a quadratic polynomial. In this case, the degree of the numerator is less than the degree of the denominator. When the degree of the numerator is smaller, the horizontal asymptote is y = 0.
So, the horizontal asymptote for the second function is y = 0.
To summarize:
First Rational Function:
- Vertical asymptote: x = 1/2
- Horizontal asymptote: y = 1/2
Second Rational Function:
- No vertical asymptotes
- Horizontal asymptote: y = 0