Determine the holes, vertical asymptotes and horizontal asymptotes of the
rational function. y=3x^2+10x-8/x^2+7x+12
Can you also show your work. I need to do corrections. I have a 68 I just need to get to a 70. This would mean so much to me if anybody could help. I'm begging here~~~
I am sure you meant
y= (3x^2+10x-8)/(x^2+7x+12 )
= (x+4)(3x-2)/(x+3)(x+4)
= (3x-2)/(x+3)
so you have a vertical asymptotes at x = -3
since we cancelled (x+4)/(x+4) and that would be 0/0 if x = -4
we have a hole at x = -4 , y = -14/-1 or 14, hole at (-4,14)
since the numerator is degree 3 and the denominator of degree 2, there is no vertical asymptote but the functions approaches y = 3x
confirmation by Wolfram:
http://www.wolframalpha.com/input/?i=plot+y%3D+(3x%5E2%2B10x-8)%2F(x%5E2%2B7x%2B12+)
To find the holes, vertical asymptotes, and horizontal asymptotes of a rational function, we need to analyze the numerator and denominator separately, and look for certain patterns.
1. Holes:
Holes occur when a factor in the numerator cancels out with the same factor in the denominator. To find holes, set the denominator equal to zero, and solve for x. Once you have the value(s) for x, substitute them back into the simplified function to get the y-value(s) of the holes.
2. Vertical Asymptotes:
Vertical asymptotes occur when the denominator equals zero, but the numerator doesn't cancel out that factor. To find vertical asymptotes, set the denominator equal to zero and solve for x. These x-values will be the vertical asymptotes.
3. Horizontal Asymptotes:
Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, compare the degrees (exponents) of the highest power of x in the numerator and denominator.
- If the degree in the numerator is greater than the degree in the denominator, there is no horizontal asymptote.
- If the degree in the denominator is greater than the degree in the numerator, the horizontal asymptote is y = 0.
- If the degrees are equal, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and denominator.
Now let's apply these steps to the given rational function: y = (3x^2 + 10x - 8) / (x^2 + 7x + 12):
Step 1: Factor the numerator and the denominator:
Numerator: 3x^2 + 10x - 8 = (3x - 2)(x + 4)
Denominator: x^2 + 7x + 12 = (x + 3)(x + 4)
Step 2: Identify the holes:
In this case, there are no holes because no factor in the numerator cancels out with any factor in the denominator.
Step 3: Find the vertical asymptotes:
Set the denominator equal to zero:
x + 3 = 0 => x = -3
Vertical asymptote: x = -3
Step 4: Determine the horizontal asymptote:
Since the degree of the numerator (2) is equal to the degree of the denominator (2), we can find the horizontal asymptote by comparing the leading coefficients (the coefficients of the highest power terms).
Leading coefficient of numerator: 3
Leading coefficient of denominator: 1
Therefore, the horizontal asymptote is y = 3/1 = 3.
To summarize:
- There are no holes.
- The vertical asymptote is x = -3.
- The horizontal asymptote is y = 3.
By following these steps and showing your work, you can verify the correct answers and make necessary corrections. Good luck on your corrections!