To graph the asymptotes of the rational function
[tex]\[ f(x) = \frac{x^2 + 2x - 1}{x + 3} ,\][/tex]
we need to find both the vertical and horizontal asymptotes.
### Finding the Vertical Asymptotes
Vertical asymptotes occur where the denominator is equal to zero, as long as the numerator is not also zero at those points.
For the given function, the denominator is [tex]\( x + 3 \)[/tex].
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x + 3 = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
Therefore, there is a vertical asymptote at
[tex]\[ x = -3. \][/tex]
### Finding the Horizontal Asymptote
The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.
- The degree of the numerator [tex]\( x^2 + 2x - 1 \)[/tex] is [tex]\( 2 \)[/tex].
- The degree of the denominator [tex]\( x + 3 \)[/tex] is [tex]\( 1 \)[/tex].
Since the degree of the numerator is greater than the degree of the denominator (2 > 1), there is no horizontal asymptote for this function.
### Summary of the Asymptotes
The function has:
- A vertical asymptote at [tex]\( x = -3 \)[/tex].
- No horizontal asymptotes.
### Graphing the Asymptotes
To graph the function along with its asymptotes:
1. Vertical Asymptote: Draw a dashed vertical line at [tex]\( x = -3 \)[/tex]. This represents the vertical asymptote.
2. Horizontal Asymptote: Since there is no horizontal asymptote, no dashed horizontal line is needed.
These steps visually indicate the behavior of the function as it approaches the asymptote lines. For [tex]\( x \)[/tex] approaching [tex]\(-3\)[/tex], the function [tex]\( f(x) \)[/tex] behaves such that it tends towards positive or negative infinity.
Note that this analysis does not include any oblique asymptotes, which are not requested in this problem but could be investigated if necessary.
