To determine which equation describes the function [tex]\( y = x^3 \)[/tex] being vertically stretched by a factor of 4, we need to understand what it means to vertically stretch a function.
A vertical stretch involves multiplying the output of the function by a constant factor.
Here's the step-by-step process:
1. Identify the parent function: The parent function given is [tex]\( y = x^3 \)[/tex].
2. Understand the effect of a vertical stretch: Vertically stretching a function [tex]\( f(x) \)[/tex] by a factor of [tex]\( k \)[/tex] involves multiplying the function by [tex]\( k \)[/tex]. This can be written as [tex]\( y = k \cdot f(x) \)[/tex].
3. Apply the vertical stretch: Given that the factor is 4 and the parent function is [tex]\( y = x^3 \)[/tex], we multiply the output of the parent function by 4:
[tex]\[
y = 4 \cdot x^3
\][/tex]
Therefore, we get the new function:
[tex]\[
y = 4x^3
\][/tex]
4. Identify the equation from the choices: Now, we match this transformed function [tex]\( y = 4x^3 \)[/tex] with one of the given options.
The options are:
1. [tex]\( y = x^3 + 4 \)[/tex]
2. [tex]\( y = (x + 4)^3 \)[/tex]
3. [tex]\( y = 4x^3 \)[/tex]
4. [tex]\( y = x^4 \)[/tex]
The correct equation matching our transformed function is:
[tex]\[
y = 4x^3
\][/tex]
Therefore, the correct answer is the third option:
[tex]\[
y = 4x^3
\][/tex]
