Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=7 x e^{-0.5 x} \) Find the \( y \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The y-intercept(s) is/are at \( y=0 \) (Type an integer or a decimal Use a comma to separate answers as needed.) B. There are no \( y \)-intercepts. Find any horizontal asymptotes of \( f(x) \), Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is C. There are no horizontal asymptotes. (Type equations.) C. (The bottom asymptote is
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The Deep Dive
To find the \( y \)-intercept of \( f(x) = 7x e^{-0.5x} \), we evaluate \( f(0) \). Plugging in \( x = 0 \), we get \( f(0) = 7 \cdot 0 \cdot e^0 = 0 \). Therefore, the \( y \)-intercept is at \( y = 0 \). For horizontal asymptotes, we analyze the behavior of \( f(x) \) as \( x \) approaches infinity. As \( x \) increases, the term \( e^{-0.5x} \) approaches 0 faster than \( 7x \) grows, leading \( f(x) \) towards 0. Hence, there is one horizontal asymptote at \( y = 0 \). So, the correct choices are: - A. The y-intercept(s) is/are at \( y=0 \) - A. The function has one horizontal asymptote, \( y=0 \)
