how would you find the horizontal and vertical asymptote of y=x+2/x-3
and
y=-3x+2/-4x-5?
thanks
To find the horizontal and vertical asymptotes of a rational function, such as the ones you provided, you can follow a few steps:
1. Determine the degree of the numerator and the denominator of the rational function:
- The degree of the numerator is the highest power of the variable x in the numerator.
- The degree of the denominator is the highest power of the variable x in the denominator.
2. Horizontal Asymptote:
- If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0.
- If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
- If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.
3. Vertical Asymptote:
- Set the denominator equal to zero and solve for x. The roots are the values where vertical asymptotes occur.
- If the equation has a factor that occurs multiple times, the vertical asymptote occurs at that root only once.
Let's now solve the two equations you provided step-by-step:
1. For y = x + 2 / (x - 3):
- The numerator has degree 1, and the denominator also has degree 1.
2. Horizontal Asymptote:
- Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients which is (1/1) = 1.
3. Vertical Asymptote:
- Set the denominator x - 3 equal to zero and solve: x - 3 = 0 => x = 3.
- Therefore, the vertical asymptote is x = 3.
So, for y = x + 2 / (x - 3), the horizontal asymptote is y = 1, and the vertical asymptote is x = 3.
Now, let's solve the second equation:
1. For y = -3x + 2 / (-4x - 5):
- The numerator has degree 1, and the denominator also has degree 1.
2. Horizontal Asymptote:
- Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients which is (-3/-4) = 3/4.
3. Vertical Asymptote:
- Set the denominator -4x - 5 equal to zero and solve: -4x - 5 = 0 => -4x = 5 => x = -5/4.
- Therefore, the vertical asymptote is x = -5/4.
So, for y = -3x + 2 / (-4x - 5), the horizontal asymptote is y = 3/4, and the vertical asymptote is x = -5/4.