To find the horizontal and vertical asymptotes of the curve given by:

$y = \frac{2x^2 + x - 1}{x^2 + x - 72}$

Vertical Asymptotes:

Vertical asymptotes occur when the denominator is zero, and the numerator is not zero at those points. To find the vertical asymptotes, set the denominator equal to zero and solve for $x$:

$x^2 + x - 72 = 0$

We can factor this quadratic equation:

$(x - 8)(x + 9) = 0$

Setting each factor equal to zero:

$x - 8 = 0 \quad \text{or} \quad x + 9 = 0$

Thus, the vertical asymptotes are:

$x = 8, \quad x = -9$

Horizontal Asymptotes:

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

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Here, the degree of both the numerator and denominator is 2 (since both have $x^2$ as their highest degree term). The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. So, the horizontal asymptote is:

$y = \frac{2}{1} = 2$

Final Answer:

• Horizontal Asymptote: $y = 2$
• Vertical Asymptotes: $x = 8, -9$

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Would you like further explanation or details on any part of this process?

Here are some related questions you might find helpful:

1. How do you find the behavior of a function near its vertical asymptote?
2. What happens if the degree of the numerator is less than the degree of the denominator in terms of horizontal asymptotes?
3. Can a function have both vertical and horizontal asymptotes?
4. How do you determine slant asymptotes, and when do they exist?
5. How do you check the correctness of your asymptotes using limits?

Tip: To find vertical asymptotes, always check for points where the denominator equals zero but the numerator doesn’t also equal zero at the same points.