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 given graph is of the parent function ( y=\sqrt{x} ), with key points at ( (0,0) ), ( (1,1) ), ( (4,2) ), ( (9,3) ) (since ( \sqrt{0}=0 ), ( \sqrt{1}=1 ), ( \sqrt{4}=2 ), ( \sqrt{9}=3 )).

Step2: Analyze the transformation

The given function is ( y=\frac{1}{3}\sqrt{x} ), which is a vertical compression of the parent function ( y = \sqrt{x} ) by a factor of ( \frac{1}{3} ). For a vertical compression by a factor of ( a ) (where ( 0 < a < 1 )), we multiply the ( y )-coordinates of the parent function's points by ( a ).

Step3: Transform the key points

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

Step4: Graph the transformed function

Plot the transformed points ( (0,0) ), ( (1,\frac{1}{3}) ), ( (4,\frac{2}{3}) ), ( (9,1) ) and draw a smooth curve through them, which will be a vertically compressed version of the parent square - root function.

Answer:

To graph ( y = \frac{1}{3}\sqrt{x} ), we vertically compress the graph of ( y=\sqrt{x} ) (parent function) by a factor of ( \frac{1}{3} ). The key points of the parent function ( (0,0) ), ( (1,1) ), ( (4,2) ), ( (9,3) ) transform to ( (0,0) ), ( (1,\frac{1}{3}) ), ( (4,\frac{2}{3}) ), ( (9,1) ) respectively. Plot these points and draw a smooth curve through them to get the graph of ( y=\frac{1}{3}\sqrt{x} ).