Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{y}=\mathrm{l}(\mathrm{x}) \). \[ f(x)=2 x(x-3)^{3} \] What is the domain of the function? Select the correct choice below and, if necessary, fill in the answer box to complele your choice. The domain is \( \square \) (Type your answer in inferval notation. Use a comma to separate answers as needed.) What is the \( y \)-inlercept? Select the correct choice telow and, if necessary, fill in the answer box to complele your choce. A. The \( y \)-intercept is \( y= \) \( \square \) . B. There is no \( y \)-intercept What is/are the x-intercepl(s)? Select tho correct choice below and, if necessary. Iill in the answer box to complele your choice. A. The \( x \)-intercept(s) islare \( x= \) \( \square \) . (Simplify your answer. Use a comma to separate answers as needed.) B. There is no \( x \)-intercept.

**Step 1. Determine the Domain** \( f(x)=2x(x-3)^{3} \) is a polynomial function. Polynomial functions are defined for all real numbers. Therefore, the domain is \[ (-\infty,\infty) \] **Step 2. Find the \( y \)-intercept** The \( y \)-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \[ f(0)=2(0)(0-3)^{3} = 0 \] Thus, the \( y \)-intercept is at \( y=0 \). **Step 3. Find the \( x \)-intercepts** The \( x \)-intercepts are found by setting \( f(x)=0 \): \[ 2x(x-3)^{3}=0 \] This equation is satisfied when either factor is zero: 1. \( 2x=0 \) \(\Rightarrow x=0\) 2. \( (x-3)^{3}=0 \) \(\Rightarrow x-3=0 \) \(\Rightarrow x=3\) Thus, the \( x \)-intercepts are at \( x=0 \) and \( x=3 \). **Step 4. Summary of Graphing Information** - **Domain:** \((-\infty, \infty)\) - **\( y \)-intercept:** \( (0,0) \) - **\( x \)-intercepts:** \( (0,0) \) and \( (3,0) \) The function \( f(x)=2x(x-3)^{3} \) is a polynomial with degree \( 1+3=4 \) (even degree) and a positive leading coefficient. This tells us: - Both ends of the graph tend upward as \( x \to \pm\infty \). - The root at \( x=0 \) is simple (odd multiplicity one) so the graph crosses the \( x \)-axis at \( x=0 \). - The root at \( x=3 \) has multiplicity 3. Since the multiplicity is odd, the graph crosses the \( x \)-axis at \( x=3 \) as well, but with a point of inflection (a flattening) near this intercept. **Final Answers:** - **Domain:** \((-\infty, \infty)\) - **\( y \)-intercept:** \( y=0 \) - **\( x \)-intercepts:** \( x=0,\,3 \)