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 is \( y = \sqrt{x} \), which has a vertex at \((0,0)\) and passes through \((1,1)\), \((4,2)\), \((9,3)\).

Step2: Analyze Horizontal Shift

For \( y = \sqrt{x - 1} \), the \( x - 1 \) inside the square root means a horizontal shift. The rule for horizontal shifts is: if \( y = f(x - h) \), shift right by \( h \) units. Here, \( h = 1 \), so shift the parent graph right 1 unit. Now the vertex is at \((1,0)\), and points like \((1 + 1,1)=(2,1)\), \((1 + 4,2)=(5,2)\), \((1 + 9,3)=(10,3)\) are on \( y = \sqrt{x - 1} \).

Step3: Analyze Vertical Shift

For \( y = \sqrt{x - 1}-3 \), the \(-3\) outside the square root is a vertical shift. The rule for vertical shifts is: if \( y = f(x)-k \), shift down by \( k \) units. Here, \( k = 3 \), so shift the graph of \( y = \sqrt{x - 1} \) down 3 units. Now the vertex is at \((1,0 - 3)=(1,-3)\), and the points become \((2,1 - 3)=(2,-2)\), \((5,2 - 3)=(5,-1)\), \((10,3 - 3)=(10,0)\).

Step4: Plot the Transformed Graph

Using the transformed vertex \((1,-3)\) and the transformed points \((2,-2)\), \((5,-1)\), \((10,0)\), we can sketch the graph of \( y=\sqrt{x - 1}-3 \). The graph should have the same shape as \( y = \sqrt{x} \) but shifted right 1 and down 3.

Answer:

To graph \( y=\sqrt{x - 1}-3 \):

1. Start with the parent function \( y = \sqrt{x} \) (vertex \((0,0)\), passes through \((1,1)\), \((4,2)\), \((9,3)\)).
2. Shift right 1 unit (for \( x - 1 \)): Vertex becomes \((1,0)\), points become \((2,1)\), \((5,2)\), \((10,3)\).
3. Shift down 3 units (for \(-3\)): Vertex becomes \((1,-3)\), points become \((2,-2)\), \((5,-1)\), \((10,0)\).
4. Plot these points and draw the curve (same shape as \( y = \sqrt{x} \), shifted right 1, down 3).

(Note: The graph should have a vertex at \((1, -3)\) and pass through \((2, -2)\), \((5, -1)\), \((10, 0)\), curving upward to the right.)