13. Sketch: \( R(x)=\frac{2 x-4}{x-4} \). Graph any asymptotes (dashed) and intercepts i) Find \( x \) and \( y \) intercepts: ii) Vertical asymptote: iii) Horizontal asymptote: iv) Determine the behavior of the graph of \( R \) on either side of the vertical asymptote: v) Sketch detailed graph.

It seems that I am unable to create a visual graph directly. However, I can guide you on how to sketch it based on the information we have gathered. ### Steps to Sketch the Graph of \( R(x) = \frac{2x - 4}{x - 4} \): 1. **Draw the Axes:** - Draw the horizontal axis (x-axis) and the vertical axis (y-axis). 2. **Plot the Intercepts:** - Mark the \( x \)-intercept at \( (2, 0) \). - Mark the \( y \)-intercept at \( (0, 1) \). 3. **Draw the Asymptotes:** - Draw a dashed vertical line at \( x = 4 \) (vertical asymptote). - Draw a dashed horizontal line at \( y = 2 \) (horizontal asymptote). 4. **Behavior Near the Asymptotes:** - As \( x \) approaches 4 from the left, the graph goes down towards \( -\infty \). - As \( x \) approaches 4 from the right, the graph goes up towards \( +\infty \). 5. **Sketch the Curve:** - For \( x < 4 \), the graph will be below the horizontal asymptote \( y = 2 \) and will cross the \( x \)-axis at \( (2, 0) \). - For \( x > 4 \), the graph will rise above the horizontal asymptote \( y = 2 \). 6. **Final Touches:** - Ensure the graph approaches the horizontal asymptote \( y = 2 \) as \( x \) goes to \( \pm \infty \). ### Summary of the Graph: - The graph will have a \( x \)-intercept at \( (2, 0) \) and a \( y \)-intercept at \( (0, 1) \). - It will have a vertical asymptote at \( x = 4 \) and a horizontal asymptote at \( y = 2 \). - The graph will show a clear transition from \( -\infty \) to \( +\infty \) at the vertical asymptote. By following these steps, you should be able to create a detailed sketch of the function \( R(x) \). If you have graphing software or a graphing calculator, you can also input the function directly to visualize it.