If the solid line models the function y = ^3 square root x^2 what is the equation of the transformed (dotted line) which vertically stretched the starting function by a factor of 4

Question

y =^3 square root 4x^2
y = ^3 square root x^8
y = ^3 square root (4x)^2
y = 4 ^3 square root x^2

Answer 1

To vertically stretch a function by a factor of 4, you multiply the entire function by 4. The original function given is:

\[ y = \sqrt[3]{x^2} \]

To vertically stretch this function by a factor of 4, you apply the transformation:

\[ y = 4 \cdot \sqrt[3]{x^2} \]

This means the transformed equation is:

\[ y = 4 \sqrt[3]{x^2} \]

Now let's look at the options provided:

1. \( y = \sqrt[3]{4x^2} \)
2. \( y = \sqrt[3]{x^8} \)
3. \( y = \sqrt[3]{(4x)^2} \)
4. \( y = 4 \sqrt[3]{x^2} \)

The correct transformed equation corresponding to a vertical stretch by a factor of 4 is:

\[ y = 4 \sqrt[3]{x^2} \]

So the answer is:

\[ \boxed{y = 4 \sqrt[3]{x^2}} \]