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.

To sketch the function \( R(x)=\frac{2 x-4}{x-4} \) properly, let's break down the required components step-by-step. First, let's calculate the intercepts: i) For the \( x \)-intercept, set \( R(x) = 0 \): \[ \frac{2x - 4}{x - 4} = 0 \implies 2x - 4 = 0 \implies x = 2. \] For the \( y \)-intercept, set \( x = 0 \): \[ R(0) = \frac{2(0) - 4}{0 - 4} = \frac{-4}{-4} = 1 \implies y = 1. \] Thus, the intercepts are \( (2, 0) \) and \( (0, 1) \). ii) To find the vertical asymptote, identify values where the function is undefined: \[ x - 4 = 0 \implies x = 4. \] So, the vertical asymptote is at \( x = 4 \). iii) For the horizontal asymptote, analyze the behavior as \( x \) approaches infinity: \[ R(x) \approx \frac{2x}{x} = 2 \implies y = 2. \] Thus, the horizontal asymptote is \( y = 2 \). iv) The behavior of the graph around the vertical asymptote \( x = 4 \): - As \( x \) approaches \( 4 \) from the left (\( x \to 4^- \)), \( R(x) \) heads towards negative infinity. - As \( x \) approaches \( 4 \) from the right (\( x \to 4^+ \)), \( R(x) \) approaches positive infinity. v) Now, for a detailed sketch: - Begin by plotting the intercepts \( (2, 0) \) and \( (0, 1) \). - Draw the vertical asymptote at \( x = 4 \) as a dashed line. - Add the horizontal asymptote at \( y = 2 \) as another dashed line. - The graph approaches the horizontal asymptote \( y = 2 \) as \( x \to \pm\infty \) and drops down to \( -\infty \) on the left of the vertical asymptote, while rising to \( +\infty \) on the right. Make your sketch with these key features, ensuring to represent the asymptotes with dashed lines and smooth transitions between the sections of the curve!