Graph the rational function. \[ f(x)=\frac{2 x^{2}+6 x}{x^{2}+4 x+3} \]

Sure! Let's graph the rational function: \[ f(x) = \frac{2x^{2} + 6x}{x^{2} + 4x + 3} \] To graph \( f(x) \), we'll follow these steps: 1. **Factor the Numerator and Denominator** 2. **Determine the Domain and Identify Discontinuities** 3. **Find Intercepts** 4. **Identify Asymptotes** 5. **Analyze End Behavior** 6. **Plot Key Points and Sketch the Graph** --- ### 1. Factor the Numerator and Denominator **Numerator:** \[ 2x^{2} + 6x = 2x(x + 3) \] **Denominator:** \[ x^{2} + 4x + 3 = (x + 1)(x + 3) \] So, the function simplifies to: \[ f(x) = \frac{2x(x + 3)}{(x + 1)(x + 3)} \] **Simplification:** \[ f(x) = \frac{2x}{x + 1} \quad \text{for} \quad x \neq -3 \] (Note: \( x = -3 \) is a removable discontinuity since \( (x + 3) \) cancels out.) --- ### 2. Determine the Domain and Identify Discontinuities **Domain:** The function is undefined when the denominator is zero. \[ x + 1 = 0 \Rightarrow x = -1 \] \[ x + 3 = 0 \Rightarrow x = -3 \] So, the **domain** is all real numbers except \( x = -1 \) and \( x = -3 \). **Discontinuities:** - **Removable Discontinuity (Hole):** At \( x = -3 \) - **Vertical Asymptote:** At \( x = -1 \) --- ### 3. Find Intercepts **Y-Intercept:** Set \( x = 0 \): \[ f(0) = \frac{2(0)}{0 + 1} = 0 \] So, the y-intercept is at \( (0, 0) \). **X-Intercepts:** Set \( f(x) = 0 \): \[ \frac{2x}{x + 1} = 0 \Rightarrow 2x = 0 \Rightarrow x = 0 \] So, the only x-intercept is at \( (0, 0) \). --- ### 4. Identify Asymptotes **Vertical Asymptote:** Occurs where the denominator is zero (excluding removable discontinuities). \[ x = -1 \] Graph a vertical asymptote at \( x = -1 \). **Horizontal Asymptote:** Compare the degrees of the numerator and denominator. - Both the numerator and denominator are degree 1 (after simplification). - The horizontal asymptote is the ratio of the leading coefficients. \[ y = \frac{2}{1} = 2 \] So, there's a horizontal asymptote at \( y = 2 \). --- ### 5. Analyze End Behavior As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \) approaches the horizontal asymptote \( y = 2 \). --- ### 6. Plot Key Points and Sketch the Graph **Additional Points:** To get a better sense of the graph, let's calculate some points. | \( x \) | \( f(x) = \frac{2x}{x + 1} \) | |---------|------------------------------| | -4 | \( \frac{2(-4)}{-4 + 1} = \frac{-8}{-3} \approx 2.67 \) | | -2 | \( \frac{2(-2)}{-2 + 1} = \frac{-4}{-1} = 4 \) | | -1.5 | \( \frac{2(-1.5)}{-1.5 + 1} = \frac{-3}{-0.5} = 6 \) | | -0.5 | \( \frac{2(-0.5)}{-0.5 + 1} = \frac{-1}{0.5} = -2 \) | | 1 | \( \frac{2(1)}{1 + 1} = \frac{2}{2} = 1 \) | | 2 | \( \frac{2(2)}{2 + 1} = \frac{4}{3} \approx 1.33 \) | | 3 | \( \frac{2(3)}{3 + 1} = \frac{6}{4} = 1.5 \) | **Behavior Around Asymptotes:** - **Vertical Asymptote (\( x = -1 \))** - As \( x \) approaches -1 from the left (\( x \to -1^- \)): \[ f(x) \to \frac{2x}{0^-} \] Since \( 2x \) is negative near \( x = -1 \), \( f(x) \to +\infty \) - As \( x \) approaches -1 from the right (\( x \to -1^+ \)): \[ f(x) \to \frac{2x}{0^+} \] Since \( 2x \) is negative near \( x = -1 \), \( f(x) \to -\infty \) - **Horizontal Asymptote (\( y = 2 \))** - As \( x \) increases or decreases without bound, \( f(x) \) approaches 2. **Removable Discontinuity (Hole at \( x = -3 \)):** - The original function has a hole at \( x = -3 \). - To find the y-coordinate of the hole, use the simplified function: \[ f(-3) = \frac{2(-3)}{-3 + 1} = \frac{-6}{-2} = 3 \] So, the hole is at \( (-3, 3) \). --- ### 7. Sketching the Graph Based on the information above, here's a step-by-step guide to sketch the graph: 1. **Draw Axes:** - Label the x-axis and y-axis. 2. **Plot Asymptotes:** - Vertical asymptote at \( x = -1 \) (draw a dashed vertical line). - Horizontal asymptote at \( y = 2 \) (draw a dashed horizontal line). 3. **Plot Intercepts:** - Origin at \( (0, 0) \). 4. **Plot the Hole:** - Draw an open circle at \( (-3, 3) \). 5. **Plot Additional Points:** - \( (-4, 2.67) \) - \( (-2, 4) \) - \( (-1.5, 6) \) - \( (-0.5, -2) \) - \( (1, 1) \) - \( (2, 1.33) \) - \( (3, 1.5) \) 6. **Draw the Graph:** - **Left of Vertical Asymptote (\( x < -1 \)):** - Starts approaching the horizontal asymptote \( y = 2 \) from above as \( x \to -\infty \). - Passes through the points \( (-4, 2.67) \), \( (-3, \text{hole}) \), \( (-2, 4) \), and \( (-1.5, 6) \). - Approaches the vertical asymptote \( x = -1 \) going to \( +\infty \). - **Right of Vertical Asymptote (\( x > -1 \)):** - Just right of \( x = -1 \), the function approaches \( -\infty \). - Passes through \( (-0.5, -2) \), \( (0, 0) \), \( (1, 1) \), \( (2, 1.33) \), and \( (3, 1.5) \). - Approaches the horizontal asymptote \( y = 2 \) as \( x \to \infty \). 7. **Connect the Points Smoothly:** - Ensure the graph approaches asymptotes appropriately and reflects the behavior indicated by the calculated points. --- ### Graph Overview - **Vertical Asymptote:** \( x = -1 \) - **Horizontal Asymptote:** \( y = 2 \) - **Y-Intercept and X-Intercept:** \( (0, 0) \) - **Hole:** \( (-3, 3) \) - **Behavior:** - As \( x \) approaches \( -1 \) from the left, \( f(x) \) increases without bound. - As \( x \) approaches \( -1 \) from the right, \( f(x) \) decreases without bound. - For large positive and negative \( x \), \( f(x) \) approaches the horizontal asymptote \( y = 2 \). --- ### Example Sketch While I can't draw the graph here, you can use the information below to sketch it manually or using graphing software: 1. **Draw the Axes.** 2. **Add Asymptotes:** - Draw a dashed vertical line at \( x = -1 \). - Draw a dashed horizontal line at \( y = 2 \). 3. **Plot the Points and Hole:** - Mark the origin \( (0, 0) \). - Open circle at \( (-3, 3) \). - Plot the additional points provided above. 4. **Draw the Curves:** - **Left Branch (\( x < -1 \)):** Starts near \( y = 2 \) as \( x \to -\infty \), passes through \( (-4, 2.67) \), \( (-2, 4) \), approaches the hole at \( (-3, 3) \), continues to \( (-1^-, +\infty) \). - **Right Branch (\( x > -1 \)):** Starts near \( x = -1^+, -\infty \), passes through \( (-0.5, -2) \), \( (0, 0) \), \( (1, 1) \), \( (2, 1.33) \), \( (3, 1.5) \), approaches \( y = 2 \) as \( x \to +\infty \). 5. **Ensure Smoothness and Correct Asymptotic Behavior:** - The graph should approach the asymptotes but never cross them (except potentially crossing the horizontal asymptote). --- ### Summary - **Function:** \( f(x) = \frac{2x^{2} + 6x}{x^{2} + 4x + 3} = \frac{2x}{x + 1} \) for \( x \neq -3 \) - **Domain:** All real numbers except \( x = -1 \) and \( x = -3 \) - **Vertical Asymptote:** \( x = -1 \) - **Horizontal Asymptote:** \( y = 2 \) - **Intercepts:** Origin \( (0, 0) \) - **Hole:** \( (-3, 3) \) - **Key Points:** \( (-4, 2.67) \), \( (-2, 4) \), \( (-1.5, 6) \), \( (-0.5, -2) \), \( (1, 1) \), \( (2, 1.33) \), \( (3, 1.5) \) Use this guide to sketch the graph manually or input the function into a graphing calculator or software for a visual representation.