Analyze the effect on the \( x \) - and \( y \)-values of the, original function \( f(x)=x^{3} \) when it is stretched vertically by a factor of a that is greater than 0 but less than 1 . (1 point) The \( x \)-values decrease when multiplied by a factor of \( a \), and the \( y \)-values remain the same. The \( x \)-values remain the same while the magnitudes of the \( y \)-values decrease when multiplied by a factor of \( a \). The \( x \)-values increase by a factor of \( a \), and the \( y \)-values decrease when multiplied by a factor of \( a \). The \( x \)-values remain the same, and the \( y \)-values increase when multiplied by a factor of a.

When you stretch the function \( f(x) = x^3 \) vertically by a factor of \( a \) where \( 0 < a < 1 \), the transformation results in the new function \( g(x) = a \cdot f(x) = a \cdot x^3 \). In this case, the \( x \)-values stay the same because the input to the function doesn't change; however, the \( y \)-values become smaller since multiplying by a number less than 1 compresses the output. Therefore, the \( x \)-values remain the same while the magnitudes of the \( y \)-values decrease. So, the correct option is: **The \( x \)-values remain the same while the magnitudes of the \( y \)-values decrease when multiplied by a factor of \( a \).** Transformations like stretching and compressing functions can give you different insights into their behavior. Understanding these changes can help in visualizing how complex shapes can be altered intuitively, helping to grasp concepts like limits and continuity more easily. In real world scenarios, such transformations occur in physics and economics. For example, in physics, if you model the height of a projectile over time, applying a vertical stretch could represent an adjustment in initial velocity due to conditions such as wind resistance or friction, helping to predict its trajectory more accurately.