Question

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

Explanation:

Step1: Identify the parent function

The parent function of \( y = \sqrt{x + 5}+5 \) is \( y=\sqrt{x} \). The graph of \( y = \sqrt{x} \) has a starting point at \( (0,0) \) and increases slowly.

Step2: Analyze horizontal shift

For the function \( y=\sqrt{x + h} \), if \( h>0 \), the graph shifts left by \( h \) units. Here, \( h = 5 \), so the graph of \( y=\sqrt{x} \) shifts left 5 units. The starting point \( (0,0) \) of \( y=\sqrt{x} \) becomes \( (0 - 5,0)=(- 5,0) \) after the horizontal shift.

Step3: Analyze vertical shift

For the function \( y = f(x)+k \), if \( k>0 \), the graph shifts up by \( k \) units. Here, \( k = 5 \), so after the horizontal shift, we shift the graph up 5 units. The point \( (-5,0) \) becomes \( (-5,0 + 5)=(-5,5) \).

Step4: Determine other points

To graph the function, we can take other points from the parent function \( y=\sqrt{x} \), apply the transformations. For example, take \( x = 4 \) in \( y=\sqrt{x} \), we get \( y = 2 \), so the point is \( (4,2) \). After horizontal shift left 5 units, \( x=4-5=-1 \), then vertical shift up 5 units, \( y = 2 + 5=7 \), so the point is \( (-1,7) \). Take \( x = 9 \) in \( y=\sqrt{x} \), \( y = 3 \), the point is \( (9,3) \). After horizontal shift left 5 units, \( x=9 - 5=4 \), then vertical shift up 5 units, \( y=3 + 5 = 8 \), so the point is \( (4,8) \). Then we can plot these points \( (-5,5) \), \( (-1,7) \), \( (4,8) \) and draw the curve similar to the parent function but shifted.

Answer:

To graph \( y=\sqrt{x + 5}+5 \), shift the graph of \( y = \sqrt{x} \) left 5 units and up 5 units. The vertex (starting point) is at \((-5,5)\), and other points can be found by transforming points from \( y=\sqrt{x} \) (e.g., \((-1,7)\), \((4,8)\) as calculated) and drawing the square - root - shaped curve through these points.