Question

5 ( f(x) = sqrt{x - 2} - 3 )
(a) graph the function.

Explanation:

Step1: Identify the parent function

The parent function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at \( (0, 0) \), with a curve increasing slowly.

Step2: Analyze the transformation \( f(x)=\sqrt{x - 2}-3 \)

• For the horizontal shift: The \( x - 2 \) inside the square root means the graph of \( y=\sqrt{x} \) is shifted 2 units to the right. So the vertex (starting point) of the parent function \( (0, 0) \) will move to \( (2, 0) \) after the horizontal shift.
• For the vertical shift: The \( - 3 \) outside the square root means the graph is shifted 3 units down. So the point \( (2, 0) \) after the horizontal shift will move to \( (2, - 3) \)

Step3: Determine the domain and find key points

• The domain of \( f(x)=\sqrt{x - 2}-3 \) is \( x - 2\geq0\Rightarrow x\geq2 \)
• Let's find some points:
• When \( x = 2 \), \( f(2)=\sqrt{2 - 2}-3=0 - 3=-3 \), so the point is \( (2, - 3) \)
• When \( x = 3 \), \( f(3)=\sqrt{3 - 2}-3=1 - 3=-2 \), so the point is \( (3, - 2) \)
• When \( x = 6 \), \( f(6)=\sqrt{6 - 2}-3=2 - 3=-1 \), so the point is \( (6, - 1) \)
• When \( x = 11 \), \( f(11)=\sqrt{11 - 2}-3=3 - 3=0 \), so the point is \( (11, 0) \)

Step4: Plot the points and draw the graph

• Plot the points \( (2, - 3) \), \( (3, - 2) \), \( (6, - 1) \), \( (11, 0) \) on the coordinate plane.
• Since the function is a square root function, the graph will be a curve starting at \( (2, - 3) \) and increasing as \( x \) increases, passing through the other plotted points.

To graph the function:

1. Start at the point \( (2, - 3) \) (the vertex).
2. Plot the other points \( (3, - 2) \), \( (6, - 1) \), \( (11, 0) \)
3. Draw a smooth curve through these points, ensuring it follows the shape of a square - root curve (increasing, concave down) and only exists for \( x\geq2 \)

(Note: Since this is a text - based explanation for graphing, the actual graph would be drawn on the given grid with the vertex at \( (2, - 3) \) and passing through the other calculated points with the characteristic square - root curve shape.)

Answer:

The graph of \( f(x)=\sqrt{x - 2}-3 \) is a square - root curve starting at \( (2, - 3) \), passing through points like \( (3, - 2) \), \( (6, - 1) \), \( (11, 0) \) (and other points for \( x\geq2 \)) with a shape similar to \( y = \sqrt{x} \) but shifted 2 units right and 3 units down. The graph is drawn on the coordinate grid with the vertex at \( (2, - 3) \) and the curve increasing for \( x\geq2 \)