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Which of the following functions undergoes a vertical compression? Option \#1: \( f(x)=5 x^{3} \) Option \#2: \( f(x)=-\frac{3}{2} x^{3} \) Option \#3: \( f(x)=-\frac{5}{7} x^{3} \) (1 point)

Ask by Bowman Phillips. in the United States
Jan 23,2025

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Answer

Option #3: \( f(x)=-\frac{5}{7} x^{3} \) undergoes a vertical compression.

Solution

A **vertical compression** occurs when the absolute value of the coefficient multiplying the function is between 0 and 1. Let's evaluate each option: 1. \( f(x) = 5x^{3} \): The coefficient is **5**, which is greater than 1. This results in a **vertical stretch**. 2. \( f(x) = -\frac{3}{2}x^{3} \): The coefficient is **\(\frac{3}{2}\)** (or 1.5), which is also greater than 1. This is another **vertical stretch**. 3. \( f(x) = -\frac{5}{7}x^{3} \): The coefficient is **\(\frac{5}{7}\)**, which is between 0 and 1. This induces a **vertical compression**. **Answer:** Option #3: \( f(x)=-\frac{5}{7} x^{3} \)

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A vertical compression occurs when a function is multiplied by a constant that is between 0 and 1. In this case, Option #3, \( f(x)=-\frac{5}{7} x^{3} \), has a coefficient of \(-\frac{5}{7}\), which is less than 1 in absolute value, indicating that the function is compressed vertically compared to the original cubic function. Option #1 and Option #2 have coefficients that do not qualify for vertical compression. Option #1 has a coefficient of 5, which suggests vertical stretching, and Option #2 has \(-\frac{3}{2}\), which also indicates vertical stretching, even though it's negative.

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