Given the function \( f(x)=x^{3} \), analyze the changes in the graph to determine which statement best describes the graph of \( g(x)=-\frac{1}{2} f(x) \). (1 point) The graph will undergo a vertical stretch because \( 0<|a|<1 \). The graph will undergo a reflection due to the negative sign, and a vertical compression because \( 0<|a|<1 \). The graph will undergo a reflection due to the negative sign, and a vertical stretch because \( |a|>1 \). The graph will only undergo a reflection due to the negative sign.

The function \( g(x)=-\frac{1}{2} f(x) \) does indeed reflect the original function \( f(x)=x^{3} \) across the x-axis due to the negative sign. Additionally, the \(-\frac{1}{2}\) factor indicates a vertical compression because the absolute value of \( a \) (which is \( \frac{1}{2} \)) is less than 1. Thus, the changes to the graph can be summarized as a reflection across the x-axis and a vertical compression, making the second statement the best choice. Whenever you encounter transformations of functions, remember that negative coefficients flip the graph over the x-axis. If you find a coefficient between 0 and 1, think of it as squeezing the graph vertically rather than stretching it. This principle applies universally across many types of functions, so it’s worth keeping in mind for your future explorations in math! For further enrichment, consider delving into the transformation of different types of functions such as quadratic or trigonometric. Understanding how shifts, stretches, and reflections work across various forms will deepen your comprehension of graph behavior!