Let $f(x) = \frac{3x^2 - 11x + 6}{3x^2 - 13x - 10}$.

1. Finding the $x$-intercepts:

The $x$-intercepts occur when $f(x) = 0$. For a rational function, this happens when the numerator is zero, provided the denominator is not zero at those points.

Set the numerator equal to zero: $3x^2 - 11x + 6 = 0$

Factorize: $3x^2 - 11x + 6 = (3x - 2)(x - 3)$

Thus, the $x$-intercepts are: $x = \frac{2}{3}, \quad x = 3$ provided these values do not make the denominator zero.

2. Finding the $y$-intercept:

The $y$-intercept occurs when $x = 0$. Substituting $x = 0$ into $f(x)$:

$f(0) = \frac{3(0)^2 - 11(0) + 6}{3(0)^2 - 13(0) - 10} = \frac{6}{-10} = -\frac{3}{5}$

Thus, the $y$-intercept is: $(0, -\frac{3}{5})$

3. Finding the vertical asymptotes:

Vertical asymptotes occur when the denominator is zero, provided the numerator is not zero at those points.

Set the denominator equal to zero: $3x^2 - 13x - 10 = 0$

Factorize: $3x^2 - 13x - 10 = (3x + 2)(x - 5)$

Thus, the vertical asymptotes are at: $x = -\frac{2}{3}, \quad x = 5$

Summary:

• $x$-intercepts: $x = \frac{2}{3}, \; x = 3$
• $y$-intercept: $(0, -\frac{3}{5})$
• Vertical asymptotes: $x = -\frac{2}{3}, \; x = 5$

Would you like further explanation or details about any of these calculations?

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Here are 5 related questions for further exploration:

1. How do horizontal asymptotes relate to this function?
2. What is the domain of $f(x)$?
3. What happens to $f(x)$ as $x \to \infty$ or $x \to -\infty$?
4. Can this function be simplified further at certain points?
5. What is the behavior of $f(x)$ near its vertical asymptotes?

Tip: Always check whether a vertical asymptote overlaps with a hole by confirming the numerator does not simultaneously vanish at the same point!