To find the vertical asymptotes, horizontal asymptotes, and holes for the given rational function:

$f(x) = \frac{x^2 + 2x + 1}{4x^2 - x - 5}$

let’s analyze each aspect:

1. Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided that the numerator is not zero at those points.

Step:

Solve $4x^2 - x - 5 = 0$.

Using the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4}$ $x = \frac{1 \pm \sqrt{1 + 80}}{8}$ $x = \frac{1 \pm \sqrt{81}}{8}$ $x = \frac{1 \pm 9}{8}$

This gives: $x = \frac{10}{8} = \frac{5}{4} \quad \text{and} \quad x = \frac{-8}{8} = -1$

So, the vertical asymptotes are at $x = \frac{5}{4}$ and $x = -1$, unless these points are also zeros of the numerator (which would make them holes).

2. Horizontal Asymptotes

Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator.

• The numerator ($x^2 + 2x + 1$) and the denominator ($4x^2 - x - 5$) are both degree 2.
• For functions where the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients.

Thus: $y = \frac{1}{4}$

3. Holes

To find holes, check if any factors cancel between the numerator and the denominator.

Factor the numerator: $x^2 + 2x + 1 = (x + 1)^2$

And the denominator: $4x^2 - x - 5 = (4x + 5)(x - 1)$

Since no factors cancel between the numerator and the denominator, there are no holes.

Summary:

• Vertical Asymptotes: $x = \frac{5}{4}, x = -1$
• Horizontal Asymptote: $y = \frac{1}{4}$
• Holes: None (DNE)

Would you like more detailed explanations or have any questions?

Related Questions:

1. How do you find slant asymptotes for rational functions?
2. What if both numerator and denominator had a common factor?
3. Can a rational function have both vertical and horizontal asymptotes?
4. How are holes different from vertical asymptotes?
5. How can asymptotes be determined using limits?

Tip:

For functions with the same degree in numerator and denominator, remember that the horizontal asymptote is the ratio of leading coefficients.