$y=\frac{17}{x^2+5x+6}$

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02-11-2026
Mathematics

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$y=\frac{17}{x^2+5x+6}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T18:07:13-05:00

The domain is all real numbers except $x = -3$ and $x = -2$, which are the vertical asymptotes.
The horizontal asymptote is the line $y = 0$ as $x \to \pm\infty$.
The derivative is $\frac{dy}{dx} = \frac{17(-2x - 5)}{(x^2 + 5x + 6)^2}$, yielding a critical point at $x = -2.5$.
The function has a local maximum at $\boxed{(-2.5, -68)}$.

Explanation

Domain and Vertical Asymptotes

The function to analyze is given by $y = \frac{17}{x^2 + 5x + 6}$. First, we factor the quadratic expression in the denominator. Since $x^2 + 5x + 6 = (x + 2)(x + 3)$, the function can be written as $y = \frac{17}{(x + 2)(x + 3)}$. The function is undefined when the denominator is zero, so the domain of the function is all real numbers except $x = -2$ and $x = -3$. These values of $x$ correspond to the vertical asymptotes of the function.

Horizontal Asymptote

To find the horizontal asymptote, we examine the behavior of the function as $x$ approaches positive or negative infinity. As $x \to \infty$ or $x \to -\infty$, the denominator $x^2 + 5x + 6$ becomes arbitrarily large, causing the fraction $y = \frac{17}{x^2 + 5x + 6}$ to approach $0$. Thus, the horizontal asymptote is the line $y = 0$.

Local Extrema

To find the local extrema, we calculate the derivative of $y$ with respect to $x$. Using the chain rule or quotient rule, the derivative is:
\[\frac{dy}{dx} = \frac{17(-2x - 5)}{(x^2 + 5x + 6)^2}\]
Setting the derivative equal to zero, we find the critical point at $-2x - 5 = 0$, which simplifies to $x = -\frac{5}{2} = -2.5$. To find the corresponding $y$-coordinate, we substitute $x = -2.5$ into the original function:
\[y = \frac{17}{(-2.5)^2 + 5(-2.5) + 6} = \frac{17}{6.25 - 12.5 + 6} = \frac{17}{-0.25} = -68\]
Thus, the function has a local maximum at $(-2.5, -68)$. Note that since the function is negative in the interval $(-3, -2)$, this point is indeed a local maximum.

Conclusion and Summary

In summary, the function $y = \frac{17}{x^2 + 5x + 6}$ has vertical asymptotes at $x = -3$ and $x = -2$, a horizontal asymptote at $y = 0$, and a local maximum at $(-2.5, -68)$. The function is positive on $(-\infty, -3) \cup (-2, \infty)$ and negative on $(-3, -2)$. The final coordinates for the local maximum are $\boxed{(-2.5, -68)}$.

Examples

This mathematical analysis of a rational function helps in understanding how values change near singularities. In real-world engineering, such functions often model the intensity of light or sound from a point source near boundaries, or voltage levels in electrical circuits near points of interference. Understanding where these functions reach maximum or minimum levels helps engineers design systems to avoid or maximize these peaks for stability and safety.