Which equation represents a vertical dilation by a factor of 2 of the function $y=f\left(x\right)$?

• A. $y=2f\left(x\right)$
• B. $y=\frac{1}{2}f\left(x\right)$
• C. $y=f\left(x\right)+2$
• D. $y=f\left(2x\right)$

Answer: (A)

Verified step by step guidance

1. Understand that a vertical dilation (or vertical stretch) of a function by a factor of 2 means multiplying the output values of the function by 2.
2. Recall the original function is given by \(y = f(x)\), where \(f(x)\)\( represents the output for each input \)x$.
3. To apply a vertical dilation by a factor of 2, multiply the entire function \(f(x)\) by 2, resulting in the new function \(y = 2f(x)\).
4. Note that multiplying the function by 2 affects the \(y\)-values (outputs) but does not change the \(x\)-values (inputs), so the input inside the function remains \(x\).
5. Compare this with other options: multiplying inside the function (like \(f(2x)\)) changes the input and represents a horizontal transformation, and adding 2 outside the function (like \(f(x) + 2\)) represents a vertical shift, not a dilation.