https://file.aiquickdraw.com/mathbot/file/1730064446706goumnobf.jpg

To find the intercepts of the equation:

\[
y = \frac{x^2 - 9}{x^2 - 4}
\]

### 1. **x-intercept** (Set \(y = 0\)):
- For \(y = 0\), the **numerator** must be 0 (since a fraction is 0 only when the numerator is 0).
\[
x^2 - 9 = 0
\]
This is a difference of squares:
\[
(x - 3)(x + 3) = 0
\]
So, the solutions are:
\[
x = 3 \quad \text{and} \quad x = -3
\]
Thus, the **x-intercepts** are \((3, 0)\) and \((-3, 0)\).

### 2. **y-intercept** (Set \(x = 0\)):
\[
y = \frac{0^2 - 9}{0^2 - 4} = \frac{-9}{-4} = \frac{9}{4}
\]
So, the **y-intercept** is:
\[
\left(0, \frac{9}{4}\right)
\]

### 3. **Exclusions from the Domain (Undefined Points)**:
The function is undefined where the **denominator** equals 0:
\[
x^2 - 4 = 0 \quad \Rightarrow \quad (x - 2)(x + 2) = 0
\]
This gives:
\[
x = 2 \quad \text{and} \quad x = -2
\]
Thus, the function has vertical asymptotes (not intercepts) at \(x = 2\) and \(x = -2\).

---

### **Answer:**
The intercepts are:
\[
(3, 0), (-3, 0), \left(0, \frac{9}{4}\right)
\] 

Would you like a further breakdown of these calculations?

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Here are 5 related questions for deeper exploration:
1. How do you determine if a point is a vertical asymptote or a hole in the graph?
2. How does a difference of squares factorization work?
3. What does the graph of this rational function look like near the asymptotes?
4. What are horizontal asymptotes, and does this function have one?
5. How do vertical and horizontal asymptotes affect the shape of the graph?

**Tip:** Always check for domain restrictions when solving rational equations to ensure you don't miss any undefined points!

Find the intercepts of the graph of the equation y = (x^2 - 9) / (x^2 - 4).

Step-by-step solution to finding the x- and y-intercepts of the rational function y = (x^2 - 9) / (x^2 - 4). This problem covers key algebraic concepts, including difference of squares and domain restrictions. Suitable for high school students.

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