Sketch a rational function subject to the given conditions. Horizontal asymptote: \( y=2 \) Vertical asymptote: \( x=1 \) \( y \)-intercept: \( (0,3) \) \( x \)-intercept: \( \left(\frac{3}{2}, 0\right) \)

To sketch a rational function that meets the specified conditions, consider a function of the form \( f(x) = \frac{a(x - \frac{3}{2})}{x - 1} \), where \( a \) will be adjusted to ensure the horizontal asymptote and \( y \)-intercept align with the requirements. Given that there’s a horizontal asymptote at \( y = 2 \), the leading coefficients of the numerator and denominator should be \( 2 \) and \( 1 \) respectively. So you can structure the function as: \[ f(x) = \frac{2(x - \frac{3}{2})}{x - 1} \] Next, simplify this to: \[ f(x) = \frac{2x - 3}{x - 1} \] Now, check the key points: - **Horizontal asymptote \( y=2 \)**: As \( x \) approaches infinity, the \( f(x) \approx \frac{2x}{x} = 2 \). - **Vertical asymptote \( x=1 \)**: \( f(x) \) becomes undefined when \( x=1 \). - **\( y \)-intercept \( (0,3) \)**: Substitute \( x=0 \): \[ f(0) = \frac{2(0) - 3}{0 - 1} = \frac{-3}{-1} = 3 \] - **\( x \)-intercept \( \left(\frac{3}{2}, 0\right) \)**: Set the numerator equal to zero: \[ 2x - 3 = 0 \rightarrow x = \frac{3}{2} \] Now, plot these key features on a graph to visualize the function, keeping in mind the asymptotic behavior near \( x=1 \) and the intercepts.