What is the effect on the \( y \)-intercept in the graph of \( f(x)=\sqrt[3]{x} \) when \( f(x) \) is replaced by \( f(5 x) \) ? (1 point) The \( y \)-intercept shifts up by a distance of 5. The \( y \)-intercept stretches horizontally by a factor of 5 . The \( y \)-intercept compresses horizontally by a factor of 5 . The \( y \)-intercept does not change.

When we replace \( f(x) = \sqrt[3]{x} \) with \( f(5x) = \sqrt[3]{5x} \), we need to determine the \( y \)-intercept. The \( y \)-intercept occurs when \( x = 0 \). For both functions, evaluating at \( x = 0 \) gives us \( f(0) = \sqrt[3]{0} = 0 \). Therefore, the \( y \)-intercept remains at \( 0 \), indicating that the horizontal compression does not affect the position of the intercept itself. In conclusion, the \( y \)-intercept does not change at all. It's a great reminder that while transformations can alter the appearance of the graph, the intercepts can sometimes be unaffected by certain horizontal scalings. So even though we might see a visually different function, we still have that comforting \( y \)-intercept right where we left it!