The given function is:

$k(x) = \frac{3x - 9}{x + 4}$

Step 1: Analyze the Asymptotes

Vertical Asymptotes

Vertical asymptotes occur when the denominator of the function is zero (since division by zero is undefined).

In this case, the denominator is $x + 4$. Setting this equal to zero: $x + 4 = 0 \quad \Rightarrow \quad x = -4$

Thus, there is a vertical asymptote at $x = -4$.

Horizontal Asymptotes

To find the horizontal asymptote, we need to analyze the behavior of $k(x)$ as $x$ approaches $\pm \infty$.

For rational functions of the form $\frac{ax + b}{cx + d}$, the horizontal asymptote is given by the ratio of the leading coefficients, which in this case is:

$y = \frac{3}{1} = 3$

Thus, there is a horizontal asymptote at $y = 3$.

Step 2: Find Points Near the Vertical Asymptote $x = -4$

To understand the behavior of the graph around $x = -4$, let's choose points close to this asymptote, one on the left of $x = -4$ and one on the right.

Evaluate at $x = -5$ (left of $x = -4$)

$k(-5) = \frac{3(-5) - 9}{-5 + 4} = \frac{-15 - 9}{-1} = \frac{-24}{-1} = 24$ So, $k(-5) = 24$.

Evaluate at $x = -3$ (right of $x = -4$)

$k(-3) = \frac{3(-3) - 9}{-3 + 4} = \frac{-9 - 9}{1} = \frac{-18}{1} = -18$ So, $k(-3) = -18$.

Step 3: Graph the Asymptotes and Plot Points

• Vertical asymptote at $x = -4$
• Horizontal asymptote at $y = 3$
• Points: $(-5, 24)$ and $(-3, -18)$

Sketch of the graph

• As $x \to -4^{-}$, $k(x) \to +\infty$
• As $x \to -4^{+}$, $k(x) \to -\infty$

The graph approaches these asymptotes but never crosses them.

Do you want more details or have any questions?

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5 Related Questions:

1. How does the degree of the numerator and denominator affect asymptotes?
2. What is the behavior of rational functions near vertical asymptotes?
3. How can we determine slant asymptotes if they exist?
4. Can rational functions have more than one vertical or horizontal asymptote?
5. How would a transformation affect the location of asymptotes?

Tip: When identifying vertical asymptotes, always check for common factors between the numerator and denominator, as they could cancel out potential asymptotes.