Question

a dilation is the stretching or compressing of a graph of a function. two dilations of $y = \sqrt{x}$ are shown in the graph. which of the following statements are true? choose two correct answers. $y = 0.5\sqrt{x}$ is a vertical stretch of $y = \sqrt{x}$ $y = 0.5\sqrt{x}$ is a vertical compression of $y = \sqrt{x}$ $y = 2\sqrt{x}$ is a vertical stretch of $y = \sqrt{x}$ $y = 2\sqrt{x}$ is a vertical compression of $y = \sqrt{x}$

Explanation:

Step1: Recall Vertical Dilation Rules

For a function \( y = a\sqrt{x} \), if \( |a|>1 \), it's a vertical stretch of \( y = \sqrt{x} \); if \( 0<|a|<1 \), it's a vertical compression.

Step2: Analyze \( y = 0.5\sqrt{x} \)

Here, \( a = 0.5 \), and \( 0 < 0.5 < 1 \). So, \( y = 0.5\sqrt{x} \) is a vertical compression (not stretch) of \( y = \sqrt{x} \).

Step3: Analyze \( y = 2\sqrt{x} \)

Here, \( a = 2 \), and \( |2|>1 \). So, \( y = 2\sqrt{x} \) is a vertical stretch of \( y = \sqrt{x} \).

Answer:

B. \( y = 0.5\sqrt{x} \) is a vertical compression of \( y = \sqrt{x} \)
C. \( y = 2\sqrt{x} \) is a vertical stretch of \( y = \sqrt{x} \)