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Analyze the effect on the graph when \( f(x)=x^{3} \) is replaced with \( f(x)=(b x)^{3} \) where \( b>1 \). Which option correctly describes the effect? (1 point) the graph is shifted to the left the graph is horizontally compressed the graph is horizontally stretched the graph is shifted to the right

Ask by Ruiz Wyatt. in the United States
Jan 23,2025

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The graph is horizontally compressed.

Solución

When \( f(x) = x^3 \) is replaced with \( f(x) = (b x)^3 \) where \( b > 1 \), the transformation can be analyzed as follows: \[ f(x) = (b x)^3 = b^3 x^3 \] Here, the coefficient \( b^3 \) affects the vertical scaling of the graph, making it steeper. However, focusing on the horizontal aspect: - Replacing \( x \) with \( b x \) results in a horizontal compression by a factor of \( \frac{1}{b} \). Since \( b > 1 \), the graph becomes horizontally compressed. **Correct Option:** the graph is horizontally compressed

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Bonus Knowledge

When \( f(x) = x^{3} \) is replaced with \( f(x) = (b x)^{3} \) where \( b > 1 \), the effect on the graph is that it is horizontally compressed. This happens because the input to the function is multiplied by a factor \( b \), which causes the output values to reach their corresponding values more rapidly, resulting in a “squeezed” appearance along the x-axis. So in simpler terms, if you think of it like taking a rubber band and pressing it together—while the height of the graph remains the same, its width will decrease. For instance, the transformation will make points on the graph get to their maximum faster than they would in the original function.

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