Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{3}\sqrt{x}$

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 \( y=\frac{1}{3}\sqrt{x} \).