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 remain the same while the magnitudes of the y-values decrease when multiplied by a factor of a.
• 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, and the y-values increase when multiplied by a factor of a.

Explanation:

To determine the effect of a vertical stretch on \( f(x) = x^3 \) by a factor \( a \) (where \( 0 < a < 1 \)):

• Vertical stretches/scales affect the \( y \)-values (outputs) of a function, while \( x \)-values (inputs) remain unchanged.
• For a vertical stretch by factor \( a \), the new function is \( g(x) = a \cdot f(x) = a x^3 \). Since \( 0 < a < 1 \), multiplying \( y \)-values (\( f(x) \)) by \( a \) reduces their magnitude (e.g., if \( f(x) = 8 \) and \( a = 0.5 \), \( g(x) = 4 \), which is smaller).

Now, analyze the options:

• First option: Matches the logic (x-values same, y-magnitudes decrease with \( a \) (0 < a < 1)).
• Second option: Incorrect (vertical stretch does not affect x-values).
• Third option: Incorrect (vertical stretch does not affect x-values).
• Fourth option: Incorrect (since \( a < 1 \), multiplying by \( a \) decreases \( y \)-values, not increases).

Answer:

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