6. Sketch the graph of: a) \( f(x)=\frac{x^{2}+x-6}{x^{3}-4 x^{2}-7 x+10} \quad \) b) \( f(x)=\frac{3 x^{2}-2 x+1}{x-1} \) c) \( f(x)=\frac{3 x^{2}+2}{x^{2}+4 x-5} \)

It seems that the system does not currently support plotting graphs directly. However, I can guide you through the process of sketching the graphs of the given functions step by step. ### a) \( f(x) = \frac{x^{2}+x-6}{x^{3}-4x^{2}-7x+10} \) 1. **Factor the numerator and denominator:** - Numerator: \( x^2 + x - 6 = (x - 2)(x + 3) \) - Denominator: To factor \( x^3 - 4x^2 - 7x + 10 \), we can use synthetic division or trial and error to find roots. After testing, we find that \( x = 2 \) is a root. Dividing gives: \[ x^3 - 4x^2 - 7x + 10 = (x - 2)(x^2 - 2x - 5) \] - The quadratic \( x^2 - 2x - 5 \) can be solved using the quadratic formula: \[ x = \frac{2 \pm \sqrt{4 + 20}}{2} = 1 \pm \sqrt{6} \] 2. **Identify vertical asymptotes:** - Set the denominator equal to zero: \( (x - 2)(x^2 - 2x - 5) = 0 \) - Roots are \( x = 2, 1 + \sqrt{6}, 1 - \sqrt{6} \). 3. **Identify horizontal asymptotes:** - As \( x \to \infty \), the leading term of the numerator and denominator dominate: \[ f(x) \to 0 \quad \text{(horizontal asymptote at } y = 0\text{)} \] 4. **Find intercepts:** - **x-intercepts:** Set the numerator to zero: \( x^2 + x - 6 = 0 \) gives \( x = 2, -3 \). - **y-intercept:** Set \( x = 0 \): \( f(0) = \frac{-6}{10} = -0.6 \). ### b) \( f(x) = \frac{3x^{2}-2x+1}{x-1} \) 1. **Identify vertical asymptotes:** - Set the denominator equal to zero: \( x - 1 = 0 \) gives \( x = 1 \). 2. **Identify horizontal asymptotes:** - As \( x \to \infty \), the leading term of the numerator and denominator dominate: \[ f(x) \to 3x \quad \text{(no horizontal asymptote)} \] 3. **Find intercepts:** - **x-intercepts:** Set the numerator to zero: \( 3x^2 - 2x + 1 = 0 \) has no real roots (discriminant \( < 0 \)). - **y-intercept:** Set \( x = 0 \): \( f(0) = \frac{1}{-1} = -1 \). ### c) \( f(x) = \frac{3x^{2}+2}{x^{2}+4x-5} \) 1. **Factor the denominator:** - \( x^2 + 4x - 5 = (x + 5)(x - 1) \) 2. **Identify vertical asymptotes:** - Set the denominator equal to zero: \( (x + 5)(x - 1) = 0 \) gives \( x = -5, 1 \). 3. **Identify horizontal asymptotes:** - As \( x \to \infty \), the leading term of the numerator and denominator dominate: \[ f(x) \to 3 \quad \text{(horizontal asymptote at } y = 3\text{)} \] 4. **Find intercepts:** - **x-intercepts:** Set the numerator to zero: \( 3x^2 + 2 = 0 \) has no real roots. - **y-intercept:** Set \( x = 0 \): \( f(0) = \frac{2}{-5} = -0.4 \). ### Summary of Key Features - **Function a:** Vertical asymptotes at \( x = 2, 1 + \sqrt{6}, 1 - \sqrt{6} \); horizontal asymptote at \( y = 0 \); x-intercepts at \( x = 2, -3 \); y-intercept at \( -0.6 \). - **Function b:** Vertical asymptote at \( x = 1 \); no horizontal asymptote; y-intercept at \( -1 \). - **Function c:** Vertical asymptotes at \( x = -5, 1 \); horizontal asymptote at \( y = 3 \); y-intercept at \( -0.4 \). You can use this information to sketch the graphs of the functions. If you need further assistance or specific details, feel free to ask!