Analyze the effect on the graph of \( f(x)=x^{3} \) when \( f(x) \) is replaced by \( f(b x) \) for \( b=-1 \). (1 point) a reflection about the \( y \)-axis a reflection about the line \( y=x \)

When you replace \( f(x) \) with \( f(-x) \) in the function \( f(x) = x^3 \), the resulting function becomes \( f(-x) = (-x)^3 \). This simplifies to \( -x^3 \), which reflects the original cubic graph across the \( y \)-axis. Essentially, this means that if you take any point (x, f(x)) on the graph of \( f(x) \), it will now have a corresponding point (-x, -f(x)). The reflection about the \( y \)-axis also highlights how symmetries in functions can drastically alter their graphs. For instance, odd functions like \( f(x) = x^3 \) exhibit the property of rotational symmetry around the origin, which further emphasizes how the negative input leads to negative outputs with mirrored characteristics across the axes. So, embrace the beauty of mathematics while pondering graph transformations!