graph the equation shown below by transforming the given graph of the parent function. \n$y =…

Explanation:

Step1: Identify the parent function

The parent function here is ( y = \sqrt{x} ). The graph of ( y=\sqrt{x} ) passes through points like ((0,0)), ((1,1)), ((4,2)), ((9,3)) (as seen in the given graph: when ( x = 1 ), ( y = 1 ); ( x = 4 ), ( y = 2 ); ( x = 9 ), ( y = 3 )).

Step2: Analyze the transformation

The given function is ( y=\frac{1}{2}\sqrt{x} ). This is a vertical compression of the parent function ( y = \sqrt{x} ) by a factor of ( \frac{1}{2} ). For a vertical compression by a factor of ( a ) (where ( 0 < a < 1 )), we multiply the ( y )-coordinate of each point on the parent function by ( a ).

Step3: Transform the key points

• For the point ((0,0)) on ( y = \sqrt{x} ): Multiply the ( y )-coordinate by ( \frac{1}{2} ), we get ((0, 0\times\frac{1}{2})=(0,0)).
• For the point ((1,1)) on ( y = \sqrt{x} ): Multiply the ( y )-coordinate by ( \frac{1}{2} ), we get ((1, 1\times\frac{1}{2})=(1, 0.5)).
• For the point ((4,2)) on ( y = \sqrt{x} ): Multiply the ( y )-coordinate by ( \frac{1}{2} ), we get ((4, 2\times\frac{1}{2})=(4, 1)).
• For the point ((9,3)) on ( y = \sqrt{x} ): Multiply the ( y )-coordinate by ( \frac{1}{2} ), we get ((9, 3\times\frac{1}{2})=(9, 1.5)).

Step4: Graph the transformed function

Plot the transformed points ((0,0)), ((1, 0.5)), ((4, 1)), ((9, 1.5)) and draw a smooth curve through them. This curve will be a vertically compressed version of the parent square - root function, with the same domain (( x\geq0 )) and increasing nature, but "flatter" than the parent function.

Answer:

To graph ( y = \frac{1}{2}\sqrt{x} ), vertically compress the graph of ( y=\sqrt{x} ) by a factor of ( \frac{1}{2} ). Plot points ((0,0)), ((1, 0.5)), ((4, 1)), ((9, 1.5)) (obtained by multiplying the ( y )-coordinates of the parent function's key points by ( \frac{1}{2} )) and draw a smooth curve through them.