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

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)) etc.

Step2: Analyze the transformation

The given function is ( y=\sqrt{x}-3 ). This is a vertical shift of the parent function ( y = \sqrt{x} ). The general form for a vertical shift of a function ( y = f(x) ) is ( y=f(x)+k ), where ( k ) is the vertical shift. If ( k<0 ), the graph shifts down by ( |k| ) units. Here, ( k = - 3 ), so the graph of ( y=\sqrt{x} ) will shift down 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 new point is ((0,0 - 3)=(0,-3)).
• When ( x = 1 ), ( y = 1 ) for the parent function. After shifting down 3 units, the new point is ((1,1 - 3)=(1,-2)).
• When ( x = 4 ), ( y = 2 ) for the parent function. After shifting down 3 units, the new point is ((4,2 - 3)=(4,-1)).
• When ( x = 9 ), ( y = 3 ) for the parent function. After shifting down 3 units, the new point is ((9,3 - 3)=(9,0)).

Step4: Graph the transformed function

Plot the new points ((0,-3)), ((1,-2)), ((4,-1)), ((9,0)) and draw a smooth curve through them, maintaining the shape of the square - root function (increasing, concave down) but shifted down 3 units from the parent function's graph.

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 the graph is a square - root curve shifted down 3 units.