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) $$ f(x)=\frac{3 x^{2}+2}{x^{2}+4 x-5} $$

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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!
To sketch the graphs of the given functions, identify vertical and horizontal asymptotes, find x-intercepts and y-intercepts, and plot key features accordingly.

Sketch the graph of: a) b)
c)

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a)

b)

c)

Summary of Key Features

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Sketch the graph of: a) b) c)