To find the $x$-intercept and $y$-intercept for the equation $y = \frac{x-1}{(x+3)(x-2)}$, follow these steps:

1. Finding the $y$-intercept:

The $y$-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation:

$y = \frac{0 - 1}{(0 + 3)(0 - 2)} = \frac{-1}{(3)(-2)} = \frac{-1}{-6} = \frac{1}{6}$

So, the $y$-intercept is at: $(0, \frac{1}{6})$

2. Finding the $x$-intercept:

The $x$-intercept occurs when $y = 0$. This means we need to solve for $x$ when $y = 0$.

$0 = \frac{x-1}{(x+3)(x-2)}$

For a fraction to be zero, the numerator must be zero. So, solve the numerator:

$x - 1 = 0 \implies x = 1$

Thus, the $x$-intercept is at: $(1, 0)$

Conclusion:

• $x$-intercept: $(1, 0)$
• $y$-intercept: $(0, \frac{1}{6})$

Would you like more details on any part of this process?

Related Questions:

1. How do vertical asymptotes affect the graph of this function?
2. What happens when the numerator and denominator have common factors?
3. Can you find the domain of the function?
4. How would you find the horizontal asymptote of the function?
5. What is the behavior of the function near the intercepts?

Tip:

For rational functions, always check the domain by identifying values that make the denominator zero!