Question

scaling the cubic function quick check
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 increase by a factor of a, and the y - values decrease when multiplied by a factor of a.
the x - values remain the same while the magnitudes of 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.

Explanation:

When a function \(y = f(x)\) is vertically stretched by a factor \(a\) (\(0 < a<1\)), the transformation is \(y=a\cdot f(x)\). For the function \(f(x)=x^{3}\), the new function is \(g(x)=a\cdot x^{3}\). The \(x -\) values (input values) for which we evaluate the function remain the same. The \(y -\) values (output values) are multiplied by \(a\). Since \(0 < a<1\), if \(y_1=x_1^{3}\) (for the original function \(f(x)\)) and \(y_2 = a\cdot x_1^{3}\) (for the transformed function \(g(x)\)), then \(|y_2|<|y_1|\) (the magnitude of the \(y -\) value decreases).

Answer:

The \(x -\)values remain the same while the magnitudes of the \(y -\)values decrease when multiplied by a factor of \(a\).