What is one effect of replacing \( f(x) \) with \( f(b x) \) in the function \( f(x)=x^{3} \) when \( b<0 \) ? (1 point) a reflection about the line \( y=x \) a shift to the right a reflection about the \( y \)-axis a shift to the left

Replacing \( f(x) \) with \( f(b x) \) where \( b < 0 \) effectively introduces a twist to the function. Specifically, the negative \( b \) flips the input values across the \( y \)-axis. So, for \( f(x) = x^3 \), using \( f(b x) = (-b x)^3 \) mirrors the graph over the \( y \)-axis, as all x-values become their opposites. This creates a reflective effect that can surprise anyone who doesn't expect such a sudden change! Now, let's consider how this plays out with specific examples. If we take \( b = -1 \), substituting gives us \( f(-x) = (-x)^3 = -x^3 \). Compared to the original function, it’s inverted vertically as well as horizontally! It's a fantastic lesson in how something simple like flipping a sign can lead to profound changes in a graph’s structure. Remember, a little algebra can lead to a lot of surprises!