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The image contains two rational functions, and each function is accompanied by empty boxes labeled "Zeros," "Asymptote," and "End Behavior." I will help you determine these characteristics for each function.

### Function 1: \(\frac{x^2 + 4x + 3}{x^2 - 9} = 0\)

1. **Zeros**: The zeros are found by setting the numerator equal to zero:
   \[
   x^2 + 4x + 3 = 0
   \]
   Factorizing the numerator:
   \[
   (x + 1)(x + 3) = 0
   \]
   So, the zeros are \(x = -1\) and \(x = -3\).

2. **Asymptotes**: Asymptotes are found by setting the denominator equal to zero:
   \[
   x^2 - 9 = 0
   \]
   Factorizing the denominator:
   \[
   (x - 3)(x + 3) = 0
   \]
   So, the vertical asymptotes are \(x = 3\) and \(x = -3\).

3. **End Behavior**: Determine the behavior of the function as \(x \to \infty\) or \(x \to -\infty\). Since the degrees of the numerator and denominator are equal, the end behavior is determined by the leading coefficients. Thus, as \(x \to \infty\) or \(x \to -\infty\), the function approaches \( \frac{1}{1} = 1\).

### Function 2: \(\frac{x^2 - x + 1}{(x + 1)(x - 3)(x - 7)} = 0\)

1. **Zeros**: The zeros are found by setting the numerator equal to zero:
   \[
   x^2 - x + 1 = 0
   \]
   Solving this using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
   \[
   x = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2}
   \]
   The zeros are complex: \(x = \frac{1 \pm i\sqrt{3}}{2}\).

2. **Asymptotes**: Asymptotes occur where the denominator equals zero:
   \[
   (x + 1)(x - 3)(x - 7) = 0
   \]
   So, the vertical asymptotes are \(x = -1\), \(x = 3\), and \(x = 7\).

3. **End Behavior**: The degree of the denominator is higher than the numerator, so as \(x \to \infty\) or \(x \to -\infty\), the function approaches \(0\).

Would you like further details or have any questions?

### Related Questions:
1. How do you find the horizontal asymptote for a rational function?
2. What does the end behavior tell us about the function's graph?
3. How do complex zeros affect the graph of a function?
4. What are the implications of vertical asymptotes in real-world scenarios?
5. How can you use factoring techniques to simplify finding zeros and asymptotes?

### Tip:
When identifying the asymptotes and zeros of rational functions, always factorize the expressions completely to simplify the analysis!

Find the Zeros, Asymptotes, and End Behavior for the given rational functions: \(\frac{x^2 + 4x + 3}{x^2 - 9} = 0\) and \(\frac{x^2 - x + 1}{(x + 1)(x - 3)(x - 7)} = 0\).

Learn how to analyze rational functions by finding their zeros, asymptotes, and end behavior. The functions \(\frac{x^2 + 4x + 3}{x^2 - 9} = 0\) and \(\frac{x^2 - x + 1}{(x + 1)(x - 3)(x - 7)} = 0\) are solved step-by-step, including a detailed breakdown of the relevant algebraic concepts and theorems. Perfect for students in Grades 10-12.

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