Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{2}\sqrt{x}$

Explanation:

Step1: Identify the parent function

The parent function here is \( y = \sqrt{x} \), which has a graph starting at the origin \((0,0)\) and passing through points like \((1,1)\), \((4,2)\), \((9,3)\) as seen in the given graph.

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 \( \frac{1}{2} \). A vertical compression by a factor of \( a \) (where \( 0 < a < 1 \)) transforms the point \((x,y)\) on the parent function to \((x,ay)\) on the transformed function.

Step3: Transform key points

• For the point \((0,0)\) on \( y = \sqrt{x} \), substituting into \( y=\frac{1}{2}\sqrt{x} \), we get \( (0,0) \) (since \( \frac{1}{2}\times0 = 0 \)).
• For the point \((1,1)\) on \( y = \sqrt{x} \), substituting \( x = 1 \) into \( y=\frac{1}{2}\sqrt{x} \), we get \( y=\frac{1}{2}\times1=\frac{1}{2} \), so the point becomes \((1,\frac{1}{2})\).
• For the point \((4,2)\) on \( y = \sqrt{x} \), substituting \( x = 4 \) into \( y=\frac{1}{2}\sqrt{x} \), we get \( y=\frac{1}{2}\times2 = 1 \), so the point becomes \((4,1)\).
• For the point \((9,3)\) on \( y = \sqrt{x} \), substituting \( x = 9 \) into \( y=\frac{1}{2}\sqrt{x} \), we get \( y=\frac{1}{2}\times3=\frac{3}{2} \), so the point becomes \((9,\frac{3}{2})\).

Step4: Graph the transformed function

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

Answer:

To graph \( y=\frac{1}{2}\sqrt{x} \), we vertically compress the graph of \( y = \sqrt{x} \) by a factor of \( \frac{1}{2} \). The key points are transformed as follows: \((0,0)\) remains \((0,0)\), \((1,1)\) becomes \((1,\frac{1}{2})\), \((4,2)\) becomes \((4,1)\), and \((9,3)\) becomes \((9,\frac{3}{2})\). Plot these points and draw a smooth curve through them to get the graph of \( y=\frac{1}{2}\sqrt{x} \).