Find the horizontal and vertical asymptotes of (3x^2-48)/(x^2-3x+10)
c'mon, guy, this is just Algebra II.
vertical asymptotes where the denominator is zero and the numerator is not zero.
Horizontal asymptotes where x is huge. When x is huge huge huge, only the highest powers matter. So, toss out everything else in the top and bottom and see what you have left.
Wouldn't the VA be (-4,0) and (4,0) cuz that's what I got when I factored out the numerator
x^2-3x+10 is never zero, so there are no VAs.
You found the roots, where y=0.
To find the horizontal and vertical asymptotes of a function, we need to analyze the behavior of the function as it approaches positive and negative infinity.
Vertical Asymptotes:
Vertical asymptotes occur when the denominator of the function becomes zero. Therefore, we need to find the values of x where the denominator (x^2-3x+10) equals zero.
To do that, we can set the denominator equal to zero and solve the equation:
x^2 - 3x + 10 = 0
Unfortunately, this quadratic equation cannot be factored, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In our equation, a = 1, b = -3, and c = 10.
x = (-(-3) ± √((-3)^2 - 4(1)(10))) / (2(1))
x = (3 ± √(9 - 40)) / 2
x = (3 ± √(-31)) / 2
Since we have a negative value under the square root, it means that there are no real solutions. Therefore, this function does not have any vertical asymptotes.
Horizontal Asymptotes:
To find the horizontal asymptotes, we need to determine the highest power of x in both the numerator and denominator.
In this case, the highest power of x in the numerator is 2, and the highest power of x in the denominator is also 2.
Since the highest powers are the same, we need to compare the leading coefficients (the coefficients in front of the highest power terms). In this case, the leading coefficients are both 3.
If the leading coefficients are the same, the horizontal asymptote is given by the ratio of the leading coefficients. Therefore, the horizontal asymptote is y = 3.
To summarize:
- There are no vertical asymptotes.
- The horizontal asymptote is y = 3.