1. **State the problem:** We want to analyze the rational function $$y = \frac{x^2 + 7x - 11}{x^2 + 16}$$ to find points of exclusion, vertical asymptotes, zeros (x-intercepts), and end behavior. 2. **Points of Exclusion:** These occur where the denominator is zero because division by zero is undefined. - Set denominator equal to zero: $$x^2 + 16 = 0$$ - Solve for $x$: $$x^2 = -16$$ - Since $x^2$ cannot be negative for real numbers, there are no real solutions. - Therefore, **no points of exclusion**. 3. **Vertical Asymptotes:** These occur at points where the denominator is zero and the numerator is not zero. - Since denominator has no real zeros, there are **no vertical asymptotes**. 4. **Zeros (x-intercepts):** These occur where the numerator is zero. - Set numerator equal to zero: $$x^2 + 7x - 11 = 0$$ - Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=7$, $c=-11$. - Calculate discriminant: $$\Delta = 7^2 - 4(1)(-11) = 49 + 44 = 93$$ - Find roots: $$x = \frac{-7 \pm \sqrt{93}}{2}$$ - These are the two zeros of the function. 5. **End Behavior:** For large $|x|$, the highest degree terms dominate. - Numerator highest degree term: $x^2$ - Denominator highest degree term: $x^2$ - The end behavior is the ratio of leading coefficients: $$\lim_{x \to \pm \infty} y = \frac{1}{1} = 1$$ - So, the horizontal asymptote is $y=1$. 6. **Summary:** - Points of Exclusion: None - Vertical Asymptotes: None - Zeros: $$x = \frac{-7 + \sqrt{93}}{2}, \quad x = \frac{-7 - \sqrt{93}}{2}$$ - End Behavior: Horizontal asymptote at $y=1$