Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \sqrt{x + 1} - 3$

Explanation:

Step1: Identify Parent Function

The parent function for square root functions is \( y = \sqrt{x} \), which has a graph starting at \((0,0)\) and increasing slowly.

Step2: Analyze Horizontal Shift

For the function \( y = \sqrt{x + 1} - 3 \), the \( x + 1 \) inside the square root indicates a horizontal shift. The rule for horizontal shifts is: if we have \( y=\sqrt{x - h} \), it shifts \( h \) units to the right; if \( h \) is negative (like \( x+1=\sqrt{x-(-1)} \)), it shifts \( |h| \) units to the left. So, \( y = \sqrt{x + 1} \) is the parent function \( y = \sqrt{x} \) shifted 1 unit to the left.

Step3: Analyze Vertical Shift

The \( - 3 \) outside the square root indicates a vertical shift. The rule for vertical shifts is: \( y=\sqrt{x}+k \) shifts \( k \) units up if \( k>0 \), and \( |k| \) units down if \( k < 0 \). So, \( y=\sqrt{x + 1}-3 \) is \( y=\sqrt{x + 1} \) shifted 3 units down.

Step4: Transform Key Points

1. Parent function key point: The parent function \( y = \sqrt{x} \) has a starting point at \((0,0)\), and other points like \((1,1)\), \((4,2)\), \((9,3)\).
2. After horizontal shift (1 unit left):

• For \((0,0)\): \( 0 - 1=-1 \), so new point is \((-1,0)\).
• For \((1,1)\): \( 1 - 1 = 0 \), so new point is \((0,1)\).
• For \((4,2)\): \( 4 - 1 = 3 \), so new point is \((3,2)\).
• For \((9,3)\): \( 9 - 1 = 8 \), so new point is \((8,3)\).

1. After vertical shift (3 units down):

• For \((-1,0)\): \( 0 - 3=-3 \), so new point is \((-1,-3)\).
• For \((0,1)\): \( 1 - 3=-2 \), so new point is \((0,-2)\).
• For \((3,2)\): \( 2 - 3=-1 \), so new point is \((3,-1)\).
• For \((8,3)\): \( 3 - 3 = 0 \), so new point is \((8,0)\).

Step5: Graph the Transformed Function

Plot the transformed points \((-1,-3)\), \((0,-2)\), \((3,-1)\), \((8,0)\) and draw a smooth curve through them, following the shape of the square root graph (increasing, concave down), starting at \((-1,-3)\) and passing through the other transformed points.

Answer:

To graph \( y=\sqrt{x + 1}-3 \), shift the graph of \( y = \sqrt{x} \) 1 unit left and 3 units down. Key transformed points are \((-1,-3)\), \((0,-2)\), \((3,-1)\), \((8,0)\); draw a smooth curve through these points starting at \((-1,-3)\) and increasing.