Question

what transformations of the parent function, y = \sqrt{x} is the function g(x) = 2\sqrt{x + 5}? vertical stretch and a vertical shift up 5; vertical stretch and a horizontal shift left 5; vertical compression and a horizontal shift left 5; vertical stretch and a horizontal shift right 5

Explanation:

Step1: Analyze Vertical Transformation

For a function \( y = a\sqrt{x} \), the coefficient \( a \) affects vertical stretch/compression. If \( |a|>1 \), it's a vertical stretch. Here, \( a = 2 \) (since \( g(x)=2\sqrt{x + 5} \)), so vertical stretch.

Step2: Analyze Horizontal Transformation

For a function \( y=\sqrt{x - h} \), \( h \) shifts horizontally. If we have \( y=\sqrt{x + 5}=\sqrt{x-(-5)} \), \( h=-5 \), so a shift left 5 units (because \( h<0 \) shifts left).

Answer:

B. Vertical stretch and a horizontal shift left 5