graph the equation shown below by transforming the given graph of the parent function. \n$y = \\sqrt{x + 5}…

Explanation:

Step1: Identify the parent function

The parent function of ( y = \sqrt{x + 5}+2 ) is ( y=\sqrt{x} ). The graph of ( y = \sqrt{x} ) has a vertex at ( (0,0) ) and passes through points like ( (1,1) ), ( (4,2) ), etc.

Step2: Analyze horizontal shift

For the function ( y=\sqrt{x + h} ), the horizontal shift is ( h ) units. Here, in ( y=\sqrt{x + 5} ), ( h=- 5 ), so the graph of ( y = \sqrt{x} ) is shifted 5 units to the left. So the vertex of ( y=\sqrt{x + 5} ) will be at ( (-5,0) ).

Step3: Analyze vertical shift

For the function ( y=\sqrt{x}+k ), the vertical shift is ( k ) units. Here, in ( y=\sqrt{x + 5}+2 ), ( k = 2 ), so the graph of ( y=\sqrt{x + 5} ) is shifted 2 units up. So the vertex of ( y=\sqrt{x + 5}+2 ) will be at ( (-5,2) ).

Step4: Determine key points

• For the parent function ( y=\sqrt{x} ), when ( x = 0 ), ( y = 0 ); ( x=1 ), ( y = 1 ); ( x = 4 ), ( y=2 ); ( x=9 ), ( y = 3 ).
• After horizontal shift (5 units left), the x - values become ( x=0 - 5=-5 ), ( x = 1-5=-4 ), ( x=4 - 5=-1 ), ( x=9 - 5 = 4 ).
• After vertical shift (2 units up), the y - values become ( y=0 + 2=2 ), ( y=1+2 = 3 ), ( y=2 + 2=4 ), ( y=3+2=5 ).

So the key points of ( y=\sqrt{x + 5}+2 ) are ( (-5,2) ), ( (-4,3) ), ( (-1,4) ), ( (4,5) ). To graph the function, we plot these points and draw a smooth curve (since it's a square - root function) passing through them, starting from the vertex ( (-5,2) ) and increasing as ( x ) increases.

Answer:

To graph ( y=\sqrt{x + 5}+2 ), shift the graph of ( y = \sqrt{x} ) 5 units to the left and 2 units up. The vertex is at ((-5,2)), and key points are ((-5,2)), ((-4,3)), ((-1,4)), ((4,5)) (plot these points and draw a smooth curve).