Question

begin by graphing the square root function, ( f(x) = sqrt{x} ). then, use transformations of this graph to graph the given function. ( h(x) = sqrt{-x} - 3 ) choose the correct graph below. a. image of graph b. image of graph c. image of graph d. image of graph

Explanation:

Step1: Analyze the parent function

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

Step2: Analyze the transformations for \( h(x) = \sqrt{-x} - 3 \)

• Reflection over the y - axis: The \( -x \) inside the square root reflects the graph of \( y=\sqrt{x} \) over the y - axis. So the domain of \( y = \sqrt{-x} \) is \( -x\geq0\) or \( x\leq0 \).
• Vertical shift down: The \( - 3 \) at the end of the function shifts the graph of \( y=\sqrt{-x} \) down by 3 units.

Step3: Analyze the key points

• For the parent function \( f(x)=\sqrt{x} \), when \( x = 0 \), \( y = 0 \). After reflection over the y - axis, for \( y=\sqrt{-x} \), when \( x = 0 \), \( y = 0 \). After shifting down 3 units, when \( x = 0 \), \( y=\sqrt{-0}-3=-3 \).
• Let's find another point. For \( y=\sqrt{-x}-3 \), if \( x=-1 \), then \( y=\sqrt{-(-1)}-3=1 - 3=-2 \). If \( x = - 4 \), then \( y=\sqrt{-(-4)}-3=2 - 3=-1 \). If \( x=-9 \), then \( y=\sqrt{-(-9)}-3 = 3-3=0 \).

Step4: Match with the graphs

• Graph A: Seems to be a linear - like graph, not a square - root graph.
• Graph B: The graph is in the first quadrant (domain \( x\geq0 \)) which does not match the domain \( x\leq0 \) of \( h(x) \).
• Graph C: Let's check the key points. When \( x = 0 \), \( y=-3 \)? No, the graph in C at \( x = 0 \) is not - 3.
• Graph D: The graph is in the second quadrant (domain \( x\leq0 \)) and at \( x = 0 \), \( y=-3 \) (matches our calculation), and as \( x \) decreases (more negative), \( y \) increases (since \( \sqrt{-x} \) increases as \( x \) becomes more negative), which matches the behavior of \( y=\sqrt{-x}-3 \).

Answer:

D (the graph corresponding to option D)