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 for \( y = \frac{1}{2}\sqrt{x} \) is \( y=\sqrt{x} \). The graph of \( y = \sqrt{x} \) has a domain \( x\geq0 \) and passes through points like \( (0,0) \), \( (1,1) \), \( (4,2) \), \( (9,3) \) (as seen in the given graph with blue points at, e.g., \( (1,1) \), \( (4,2) \), \( (9,3) \)).

Step2: Analyze the transformation

The function \( y=\frac{1}{2}\sqrt{x} \) 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 \( a \) (where \( 0 < a < 1 \)), we multiply the \( y \)-values of the parent function by \( a \).

Step3: Transform key points

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

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, maintaining the shape of the square - root function (increasing, concave down) but with the \( y \)-values compressed towards the \( x \)-axis.

Answer:

To graph \( y=\frac{1}{2}\sqrt{x} \):

1. Start with the parent function \( y = \sqrt{x} \) (graph with points \((0,0)\), \((1,1)\), \((4,2)\), \((9,3)\)).
2. Vertically compress the parent function by a factor of \( \frac{1}{2} \) (multiply \( y \)-coordinates of parent function points by \( \frac{1}{2} \)) to get points \((0,0)\), \((1,\frac{1}{2})\), \((4,1)\), \((9,\frac{3}{2})\).
3. Plot these new points and draw a smooth curve through them, which is the graph of \( y=\frac{1}{2}\sqrt{x} \).