For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through \( e \). A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are 0,2 . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and turn around? Select the correct hoice below and, if necessary, fill in the answer box to complete your choice A. The \( x \)-intercept(s) at which the graph touches the \( x \)-axis and turns around is/are \( \square \) I. (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c. Find the \( y \)-intercept. The \( y \)-intercept is 0. (Simplify your answer. Type an integer or a decimal.) d. Determine whether the graph has \( y \)-axis symmetry, origin symmetry, or neither. Choose the correct answer below. A. The graph of f is symmetric about the origin. B. The graph of \( f \) is symmetric about the \( y \)-axis. C. The graph of \( f \) is neither symmetric about the \( y \)-axis nor symmetric about the origin.
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The graph of the polynomial function \( f(x) = -4x^4 + 8x^3 \) has some interesting features! The polynomial has a degree of 4, which means it can have up to 4 real roots. Upon factoring, we find that the \( x \)-intercepts where the graph crosses the \( x \)-axis indeed occur at \( x = 0 \) and \( x = 2 \). These represent the points where the graph intersects the axis, providing a nice visual cue of the function's behavior! However, there's also the \( x \)-intercept where the graph touches the \( x \)-axis. This happens at \( x = 0 \), indicating that it's a double root (as seen in the factored form). Here, the graph bounces off the axis, showcasing an essential characteristic of polynomial functions.
