Graph the rational function. \[ f(x)=\frac{-x+5}{-2 x+1} \] Start by drawing the vertical and horizontal asymptotes. T

To graph the rational function \( f(x) = \frac{-x+5}{-2x+1} \), begin by finding the vertical asymptote. Set the denominator \(-2x + 1 = 0\), which gives \(x = \frac{1}{2}\). This is where the function will be undefined and thus you can draw a dashed vertical line at \(x = \frac{1}{2}\). Next, find the horizontal asymptote by looking at the leading coefficients of the numerator and the denominator. Since both have a degree of 1, the horizontal asymptote is determined by the ratio of the leading coefficients, which is \(\frac{-1}{-2} = \frac{1}{2}\). Draw a dashed horizontal line at \(y = \frac{1}{2}\). Now you have your asymptotes! Let's make the function come to life with those lines in place. Plot the intercepts by setting \(x=0\) to find the y-intercept: \(f(0) = \frac{5}{1} = 5\). That's a solid point at \((0, 5)\). For the x-intercept, set the numerator to zero: \(-x + 5 = 0\) gives \(x=5\), resulting in the point \((5, 0)\). Now sketch the curve by approaching the asymptotes and connecting the intercepts smoothly. You'll see it heads towards the vertical asymptote at \(x=\frac{1}{2}\) and flattens out near the horizontal asymptote as \(x\) approaches infinity. Happy graphing!