If the graph of $f(x)=\frac{9 x^2+36 x+41}{3 x+5}$ has an oblique asymptote at $y=3 x+k$, what is the value of $k$?

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simplyfawn simplyfawn

05-11-2026
Mathematics

Answered

If the graph of $f(x)=\frac{9 x^2+36 x+41}{3 x+5}$ has an oblique asymptote at $y=3 x+k$, what is the value of $k$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T23:45:30-05:00

Perform polynomial long division to express $f(x)$ as $3x + 7 + \frac{6}{3x + 5}$.
Identify that as $x \to \pm \infty$, the function approaches the linear equation $y = 3x + 7$.
Match this result with the given form $y = 3x + k$ to find $k$.
Final answer: $\boxed{7}$.

Explanation

Understanding the Problem and Approach

The objective is to find the value of $k$ for the oblique asymptote $y = 3x + k$ of the function $f(x) = \frac{9x^2 + 36x + 41}{3x + 5}$. An oblique asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator. We can find this by performing polynomial division of the numerator by the denominator. If we express $f(x)$ as $q(x) + \frac{r(x)}{d(x)}$, where $q(x)$ is a linear polynomial, then $y = q(x)$ represents the equation of the oblique asymptote.

Performing Polynomial Long Division

We perform long division of $9x^2 + 36x + 41$ by $3x + 5$.

Divide the leading term of the numerator ($9x^2$) by the leading term of the denominator ($3x$):

$\frac{9x^2}{3x} = 3x$.

Multiply $3x$ by the denominator ($3x + 5$):

$3x(3x + 5) = 9x^2 + 15x$.

Subtract this from the original numerator:

$(9x^2 + 36x + 41) - (9x^2 + 15x) = 21x + 41$.

Now, divide the leading term of the new expression ($21x$) by the leading term of the denominator ($3x$):

$\frac{21x}{3x} = 7$.

Multiply $7$ by the denominator ($3x + 5$):

$7(3x + 5) = 21x + 35$.

Subtract this from the previous remainder ($21x + 41$):

$(21x + 41) - (21x + 35) = 6$.

Thus, the function can be written as $f(x) = 3x + 7 + \frac{6}{3x + 5}$.

Conclusion and Identification of k

By performing the long division, we determined that $f(x) = 3x + 7 + \frac{6}{3x + 5}$. As $x \to \pm \infty$, the term $\frac{6}{3x + 5}$ approaches 0. Therefore, the function $f(x)$ approaches the linear function $y = 3x + 7$. Comparing this to the given form $y = 3x + k$, we conclude that $k = 7$.

Examples

Oblique asymptotes are vital in engineering and economics to understand the long-term behavior of a system or function. For example, if a cost function per unit is defined as a ratio of total cost (a quadratic function) to production volume (a linear function), the oblique asymptote tells us the limit of the average variable cost as production scales to very large numbers. This helps businesses predict long-term marginal costs and optimize production strategies.