Find the vertical and horizontal asymtotes of y=1/x^2+6x+9
To find the vertical and horizontal asymptotes of the given function, we need to analyze the behavior of the function as x approaches positive or negative infinity.
First, let's rewrite the given function in a more simplified form. We can rearrange the equation as follows:
y = 1/(x^2 + 6x + 9)
Next, let's determine the vertical asymptotes. Vertical asymptotes occur when the function approaches infinity or negative infinity for certain values of x.
To find the vertical asymptotes, we need to determine the values of x that make the denominator of the function equal to zero. In this case, the denominator is x^2 + 6x + 9. Setting it equal to zero and solving for x:
x^2 + 6x + 9 = 0
Using factoring or the quadratic formula, we can determine that this quadratic equation factors as:
(x + 3)(x + 3) = 0
This tells us that the only solution to the equation is x = -3. Therefore, the vertical asymptote of the function is x = -3.
Now let's move on to the horizontal asymptote. Horizontal asymptotes occur when the function approaches a certain value as x approaches positive or negative infinity.
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the function. In this case, the degree of the numerator is zero (since it's a constant), and the degree of the denominator is 2 (since it's a quadratic equation).
When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote occurs at y = 0.
Therefore, the horizontal asymptote of the function y = 1/(x^2 + 6x + 9) is y = 0.
To summarize:
- Vertical asymptote: x = -3
- Horizontal asymptote: y = 0