Sletontue graph of eain of the followins function [ d k(x)=(x+6)/(x^2)+4 ]

To sketch the graph of the function \( k(x) = \frac{x+6}{x^2+4} \), we need to analyze its key features:<br /><br />1. **Domain**: The function is defined for all real numbers because the denominator \( x^2 + 4 \) is never zero (since \( x^2 + 4 \geq 4 \) for all \( x \)).<br /><br />2. **Intercepts**:<br /> - **y-intercept**: Set \( x = 0 \):<br /> \[<br /> k(0) = \frac{0 + 6}{0^2 + 4} = \frac{6}{4} = \frac{3}{2}<br /> \]<br /> - **x-intercept**: Set \( k(x) = 0 \):<br /> \[<br /> \frac{x + 6}{x^2 + 4} = 0 \implies x + 6 = 0 \implies x = -6<br /> \]<br /><br />3. **Asymptotes**:<br /> - **Vertical asymptotes**: There are no vertical asymptotes because the denominator \( x^2 + 4 \) is never zero.<br /> - **Horizontal asymptote**: As \( x \to \pm\infty \), the degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is \( y = 0 \).<br /><br />4. **Behavior near intercepts and asymptotes**:<br /> - Near \( x = -6 \), the function crosses the x-axis.<br /> - As \( x \to \pm\infty \), \( k(x) \to 0 \).<br /><br />5. **First derivative** (for increasing/decreasing behavior):<br /> \[<br /> k'(x) = \frac{(x^2 + 4)(1) - (x + 6)(2x)}{(x^2 + 4)^2} = \frac{x^2 + 4 - 2x^2 - 12x}{(x^2 + 4)^2} = \frac{-x^2 - 12x + 4}{(x^2 + 4)^2}<br /> \]<br /> Analyze the sign of \( k'(x) \) to determine intervals of increase and decrease.<br /><br />6. **Second derivative** (for concavity and inflection points):<br /> \[<br /> k''(x) = \text{(complex expression)}<br /> \]<br /> Analyze the sign of \( k''(x) \) to determine concavity.<br /><br />By combining these analyses, you can sketch the graph of \( k(x) \). The graph will have a y-intercept at \( \left(0, \frac{3}{2}\right) \), an x-intercept at \( (-6, 0) \), and approach the horizontal asymptote \( y = 0 \) as \( x \to \pm\infty \). The function will be smooth and continuous for all \( x \).