Question

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

Explanation:

Step1: Identify the parent function

The parent function here is \( y = \sqrt{x} \). The graph of \( y=\sqrt{x} \) starts at the origin \((0,0)\) and increases slowly, passing through points like \((1,1)\), \((4,2)\), \((9,3)\) as shown in the given graph.

Step2: Analyze the transformation

The given function is \( y=\sqrt{x}-3 \). For a function of the form \( y = f(x)+k \) (or in this case \( y = f(x)-3 \) where \( f(x)=\sqrt{x} \)), the transformation is a vertical shift. If \( k>0 \), it is a shift up, if \( k < 0 \), it is a shift down. Here \( k=- 3 \), so we need to shift the graph of the parent function \( y = \sqrt{x} \) down by 3 units.

Step3: Apply the transformation to key points

• For the parent function \( y=\sqrt{x} \), when \( x = 0 \), \( y=0 \). After shifting down 3 units, the point becomes \((0,0 - 3)=(0,-3)\).
• When \( x = 1 \), \( y = 1 \). After shifting down 3 units, the point becomes \((1,1 - 3)=(1,-2)\).
• When \( x = 4 \), \( y = 2 \). After shifting down 3 units, the point becomes \((4,2 - 3)=(4,-1)\).
• When \( x=9 \), \( y = 3 \). After shifting down 3 units, the point becomes \((9,3 - 3)=(9,0)\).

To graph \( y=\sqrt{x}-3 \), we take the graph of \( y = \sqrt{x} \) and move each point on it down by 3 units. So the new graph will have the same shape as \( y=\sqrt{x} \) but will be shifted vertically downward by 3 units.

Answer:

To graph \( y=\sqrt{x}-3 \), shift the graph of the parent function \( y = \sqrt{x} \) down 3 units. The key points of the transformed graph are \((0, - 3)\), \((1,-2)\), \((4,-1)\), \((9,0)\) (and other corresponding points from the parent function shifted down), and the graph has the same shape as \( y=\sqrt{x} \) but is vertically translated downward by 3 units.