Question

y = -2 + \sqrt3{x}
identify the transformations.
reflection and down 2
reflection and stretch by 2
down 2
up 2

Explanation:

Step1: Recall transformation rules for cube root functions

The parent function of the cube root function is \( y = \sqrt[3]{x} \). For vertical shifts, if we have a function \( y=\sqrt[3]{x}+k \), when \( k>0 \), it is a shift up by \( k \) units, and when \( k < 0 \), it is a shift down by \(|k|\) units. Also, for reflections, a negative sign in front of the cube root (like \( y=-\sqrt[3]{x} \)) would be a reflection over the \( x \)-axis, but here we have \( y=- 2+\sqrt[3]{x}=\sqrt[3]{x}-2\).

Step2: Analyze the given function

The given function is \( y=\sqrt[3]{x}-2 \). Comparing with the parent function \( y = \sqrt[3]{x} \), we can see that there is a vertical shift. Since we are subtracting 2 from the cube root function, this represents a vertical shift down by 2 units. There is no reflection (because there is no negative sign in front of the cube root term) and no stretching (the coefficient of \( \sqrt[3]{x} \) is 1, not a number other than 1 or - 1 that would indicate a stretch or compression).

Answer:

C. Down 2 (assuming the third option is labeled as C, if the options are A:Reflection and down 2, B:Reflection and stretch by 2, C:Down 2, D:Up 2)