sketch graph of $$ f(x)=\frac{3 x^{2}+2}{x^{2}+4 x} $$

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To sketch the graph of the function $$ f(x) = \frac{3x^{2} + 2}{x^{2} + 4x} $$, we'll analyze its key features, including domain, asymptotes, intercepts, and critical points. Here's a step-by-step guide to help you visualize and draw the graph accurately. ### 1. **Determine the Domain** The **domain** consists of all real numbers for which the denominator is not zero. $$ x^{2} + 4x \neq 0 \implies x(x + 4) \neq 0 \implies x \neq 0 \text{ and } x \neq -4 $$ **Domain:** All real numbers except $$ x = 0 $$ and $$ x = -4 $$. ### 2. **Identify Vertical Asymptotes** Vertical asymptotes occur where the denominator is zero (and the numerator isn't zero at those points). From the domain: - $$ x = 0 $$ - $$ x = -4 $$ **Vertical Asymptotes:** $$ x = 0 $$ and $$ x = -4 $$. ### 3. **Determine the Horizontal Asymptote** For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of their leading coefficients. - **Numerator:** $$ 3x^{2} + 2 $$ (degree 2) - **Denominator:** $$ x^{2} + 4x $$ (degree 2) $$ \text{Horizontal Asymptote} = \frac{3}{1} = 3 \quad \Rightarrow \quad y = 3 $$ ### 4. **Find Intercepts** #### a. **Y-Intercept** Set $$ x = 0 $$: $$ f(0) = \frac{3(0)^2 + 2}{0^2 + 4(0)} = \frac{2}{0} $$ Undefined. **No y-intercept**. #### b. **X-Intercepts** Set $$ f(x) = 0 $$: $$ \frac{3x^{2} + 2}{x^{2} + 4x} = 0 \implies 3x^{2} + 2 = 0 $$ $$ x^{2} = -\frac{2}{3} $$ No real solutions. **No x-intercepts**. ### 5. **Determine Critical Points** Find where the first derivative $$ f'(x) $$ is zero or undefined to locate local maxima and minima. #### a. **Compute the Derivative** $$ f(x) = \frac{3x^{2} + 2}{x^{2} + 4x} $$ Using the quotient rule: $$ f'(x) = \frac{(6x)(x^{2} + 4x) - (3x^{2} + 2)(2x + 4)}{(x^{2} + 4x)^2} $$ Simplifying the numerator: $$ 6x(x^{2} + 4x) - (3x^{2} + 2)(2x + 4) = 6x^3 + 24x^2 - 6x^3 - 12x^2 - 4x - 8 = 12x^2 - 4x - 8 $$ $$ f'(x) = \frac{12x^2 - 4x - 8}{(x^{2} + 4x)^2} = \frac{4(3x^2 - x - 2)}{(x^{2} + 4x)^2} $$ #### b. **Set $$ f'(x) = 0 $$** $$ 3x^2 - x - 2 = 0 $$ Using the quadratic formula: $$ x = \frac{1 \pm \sqrt{1 + 24}}{6} = \frac{1 \pm 5}{6} $$ $$ x = 1 \quad \text{or} \quad x = -\frac{2}{3} $$ #### c. **Find Corresponding $$ y $$-Values** - **At $$ x = 1 $$:** $$ f(1) = \frac{3(1)^2 + 2}{1^2 + 4(1)} = \frac{5}{5} = 1 $$ - **At $$ x = -\frac{2}{3} $$:** $$ f\left(-\frac{2}{3}\right) = \frac{3\left(-\frac{2}{3}\right)^2 + 2}{\left(-\frac{2}{3}\right)^2 + 4\left(-\frac{2}{3}\right)} = \frac{\frac{12}{9} + \frac{18}{9}}{-\frac{20}{9}} = \frac{\frac{30}{9}}{-\frac{20}{9}} = -\frac{3}{2} $$ **Critical Points:** - $$ (1, 1) $$ — Local Maximum - $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$ — Local Minimum ### 6. **Analyze End Behavior** As $$ x \to \pm\infty $$: $$ f(x) = \frac{3x^{2} + 2}{x^{2} + 4x} \approx \frac{3x^{2}}{x^{2}} = 3 $$ **End Behavior:** The graph approaches the horizontal asymptote $$ y = 3 $$ from both sides. ### 7. **Behavior Near Vertical Asymptotes** - **At $$ x = 0 $$:** - **From the right ($$ x \to 0^+ $$):** $$ f(x) \to +\infty $$ - **From the left ($$ x \to 0^- $$):** $$ f(x) \to -\infty $$ - **At $$ x = -4 $$:** - **From the right ($$ x \to -4^+ $$):** $$ f(x) \to -\infty $$ - **From the left ($$ x \to -4^- $$):** $$ f(x) \to +\infty $$ ### 8. **Summary of Key Features** - **Domain:** $$ x \neq 0, -4 $$ - **Vertical Asymptotes:** $$ x = 0 $$, $$ x = -4 $$ - **Horizontal Asymptote:** $$ y = 3 $$ - **Intercepts:** None - **Critical Points:** $$ (1, 1) $$ and $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$ - **Behavior:** - Approaches $$ y = 3 $$ as $$ x \to \pm\infty $$ - Goes to $$ \pm\infty $$ near vertical asymptotes ### 9. **Sketching the Graph** While I can't provide a visual sketch here, follow these guidelines to draw the graph: 1. **Draw the Asymptotes:** - Draw dashed vertical lines at $$ x = 0 $$ and $$ x = -4 $$. - Draw a dashed horizontal line at $$ y = 3 $$. 2. **Plot Critical Points:** - Plot the points $$ (1, 1) $$ and $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$. 3. **Determine the Regions:** - **Region I:** $$ x < -4 $$ - **Region II:** $$ -4 < x < 0 $$ - **Region III:** $$ x > 0 $$ 4. **Analyze Each Region:** - **Region I ($$ x < -4 $$):** - The function approaches $$ y = 3 $$ from above as $$ x \to -\infty $$. - As $$ x $$ approaches $$ -4 $$ from the left, $$ f(x) \to +\infty $$. - Pass through $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$. - **Region II ($$ -4 < x < 0 $$):** - As $$ x $$ approaches $$ -4 $$ from the right, $$ f(x) \to -\infty $$. - As $$ x $$ approaches $$ 0 $$ from the left, $$ f(x) \to -\infty $$. - Pass through $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$. - **Region III ($$ x > 0 $$):** - As $$ x $$ approaches $$ 0 $$ from the right, $$ f(x) \to +\infty $$. - The function approaches $$ y = 3 $$ from below as $$ x \to +\infty $$. - Pass through $$ (1, 1) $$. 5. **Connect the Points Smoothly:** - Ensure the graph approaches the asymptotes correctly. - The function should have smooth curves passing through the critical points without crossing the vertical asymptotes. ### 10. **Final Graph Features** - **No x-intercepts or y-intercepts.** - **Local Maximum** at $$ (1, 1) $$. - **Local Minimum** at $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$. - **Symmetric Behavior** around the asymptotes with the horizontal asymptote guiding the end behavior. By following these steps and features, you can accurately sketch the graph of $$ f(x) = \frac{3x^{2} + 2}{x^{2} + 4x} $$. To sketch the graph of $$ f(x) = \frac{3x^{2} + 2}{x^{2} + 4x} $$, follow these steps: 1. **Domain:** All real numbers except $$ x = 0 $$ and $$ x = -4 $$. 2. **Vertical Asymptotes:** $$ x = 0 $$ and $$ x = -4 $$. 3. **Horizontal Asymptote:** $$ y = 3 $$. 4. **No x-intercepts or y-intercepts.** 5. **Critical Points:** - Local Maximum at $$ (1, 1) $$. - Local Minimum at $$ \left(-\frac{2}{3}, -\frac{3}{2}\right) $$. 6. **Behavior:** - Approaches $$ y = 3 $$ as $$ x \to \pm\infty $$. - Approaches $$ \pm\infty $$ near vertical asymptotes. Plot the asymptotes, critical points, and sketch the curve accordingly to visualize the graph.

sketch graph of \( f(x)=\frac{3 x^{2}+2}{x^{2}+4 x} \)

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