Select the correct answer. The graphs of rational functions \( m \) and \( n \) have the same vertical asymptotes, and both have a single \( x \)-interc Which equation could represent function \( n \) ? A. \( n(x)=5 m(x) \) B. \( n(x)=m(x)+5 \) C. \( n(x)=m(x-5) \) D. \( n(x)=m(x+5) \)

To determine which equation could represent function \( n \), let's analyze the given information: 1. **Same Vertical Asymptotes**: - Vertical asymptotes of a rational function are determined by the factors in the denominator. Since \( m \) and \( n \) have the same vertical asymptotes, their denominators must be identical. This means that any horizontal or vertical shifts would alter the position of these asymptotes. 2. **Single \( x \)-Intercept**: - Both functions have exactly one \( x \)-intercept, meaning each function crosses the \( x \)-axis at only one point. Now, let's evaluate each option: - **A. \( n(x) = 5m(x) \)** - **Effect**: This scales the function vertically by a factor of 5. - **Impact on Asymptotes**: Vertical asymptotes remain unchanged because scaling does not affect the denominator. - **Impact on \( x \)-Intercept**: The single \( x \)-intercept remains, as scaling does not change the location of the intercept. - **B. \( n(x) = m(x) + 5 \)** - **Effect**: This shifts the function vertically upward by 5 units. - **Impact on Asymptotes**: Vertical asymptotes remain unchanged. - **Impact on \( x \)-Intercept**: The \( x \)-intercept might no longer exist if the entire graph moves above the \( x \)-axis, or it might still intersect once depending on the original function. - **C. \( n(x) = m(x - 5) \)** - **Effect**: This shifts the function horizontally to the right by 5 units. - **Impact on Asymptotes**: Vertical asymptotes would shift, which contradicts the given condition. - **D. \( n(x) = m(x + 5) \)** - **Effect**: This shifts the function horizontally to the left by 5 units. - **Impact on Asymptotes**: Vertical asymptotes would shift, which contradicts the given condition. **Conclusion:** Option **A** maintains the same vertical asymptotes and preserves the single \( x \)-intercept, making it the correct choice. **Answer:** **A. \(n(x)=5 m(x)\)**