The function provided in the image is:

$f(x) = \frac{x^2 - 1}{x^2 + 4}$

This is a rational function, and we can analyze it to prepare for graphing. Let's go through some key steps to understand its behavior:

1. Domain:

Since there is no value of $x$ that will make the denominator $x^2 + 4$ equal to zero (because $x^2 + 4 > 0$ for all real $x$), the function is defined for all real numbers. Therefore, the domain is:

$\text{Domain: } (-\infty, \infty)$

2. Intercepts:

• x-intercepts: Set $f(x) = 0$, which happens when the numerator equals zero. So, solve:

  $x^2 - 1 = 0 \implies x = \pm 1$

  Therefore, the x-intercepts are at $x = 1$ and $x = -1$.

• y-intercept: Set $x = 0$ and compute $f(0)$:

  $f(0) = \frac{0^2 - 1}{0^2 + 4} = \frac{-1}{4}$

  So, the y-intercept is $(0, -\frac{1}{4})$.

3. Asymptotes:

• Vertical Asymptotes: None, since the denominator $x^2 + 4$ never equals zero.

• Horizontal Asymptote: To find the horizontal asymptote, examine the behavior of $f(x)$ as $x \to \infty$ and $x \to -\infty$. Since the degree of the numerator and the denominator are the same (both are degree 2), the horizontal asymptote is determined by the ratio of the leading coefficients:

  $\lim_{x \to \pm \infty} f(x) = \frac{1}{1} = 1$

  Therefore, there is a horizontal asymptote at $y = 1$.

4. Behavior of the Function:

• At large values of $x$, the function approaches 1 (from both positive and negative directions due to the horizontal asymptote).
• Around the x-intercepts $x = 1$ and $x = -1$, the function crosses the x-axis.

Based on this analysis, the function has the following features:

• Domain: $(-\infty, \infty)$
• x-intercepts at $x = 1$ and $x = -1$
• y-intercept at $(0, -\frac{1}{4})$
• Horizontal asymptote at $y = 1$

Would you like a more detailed step-by-step graphing of the function?

Here are 5 related questions to consider:

1. How do we confirm the behavior near the x-intercepts?
2. How can we use derivatives to check the function's increasing and decreasing intervals?
3. What is the role of symmetry in rational functions like this one?
4. How do vertical and horizontal asymptotes guide us in sketching the graph?
5. How can we interpret the function's concavity by using the second derivative?

Tip: When graphing rational functions, always examine the behavior as $x \to \pm \infty$ and near the intercepts for better accuracy!