Analyze the effect on the \( y \)-intercept in the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by af \( (x) \) and \( a \) is negative. (1 point) The \( y \)-intercept shifts up by a distance' of \( a \). The \( y \)-intercept shifts to the left by a distance of \( a \). The \( y \)-intercept shifts down by a distance of \( a \). The \( y \)-intercept does not change.

When the function \( f(x) = \sqrt[3]{x} \) is modified to \( af(x) \) with \( a \) being negative, the graph of the function reflects across the x-axis, which directly affects all y-values. The \( y \)-intercept, originally at \( (0,0) \), remains at that point for \( f(x) \) but will now be different for \( af(x) \). Since \( a \) is negative, the \( y \)-intercept moves down instead of up, thus confirming that it shifts down by a distance of \( |a| \). So, if \( a = -2 \) for example, the new y-intercept would now be \( (0,-2) \). To visualize this ain't just number crunching, think of placing a mirror under your graph. When you swap \( f(x) \) for \( af(x) \) with a negative \( a \), it's like flipping that reflection - what once was positive is now a downward dip! This marvel illustrates how transformations can radically alter graphs while also offering a fresh perspective on the math that dances along these curves.