Question

begin by graphing the square root function, f(x) = \sqrt{x}. then, use transformations of this graph to graph the given function.\\
g(x) = \frac{1}{3}\sqrt{x - 2}\\
choose the correct graph of g(x) below.\\
\bigcirc a.\\
\bigcirc b.\\
\bigcirc c.\\
\bigcirc d.

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x)=\sqrt{x} \), which has a domain \( x\geq0 \) and starts at the origin \((0,0)\), increasing slowly.

Step2: Analyze the transformation for \( \sqrt{x - 2} \)

The \( x - 2 \) inside the square root represents a horizontal shift. For a function \( \sqrt{x - h} \), it shifts the graph of \( \sqrt{x} \) to the right by \( h \) units. Here, \( h = 2 \), so the graph of \( \sqrt{x - 2} \) is the graph of \( \sqrt{x} \) shifted 2 units to the right. So the domain becomes \( x\geq2 \), and the starting point is \((2,0)\).

Step3: Analyze the transformation for \( \frac{1}{3}\sqrt{x - 2} \)

The coefficient \( \frac{1}{3} \) in front of the square root represents a vertical compression. A vertical compression by a factor of \( \frac{1}{3} \) means that each \( y \)-value of \( \sqrt{x - 2} \) is multiplied by \( \frac{1}{3} \), making the graph shorter (less steep) than the parent function.

Step4: Identify the correct graph

Now, let's check the options:

• Option A: The graph seems to start around \( x = 2 \) (due to the horizontal shift) and has a vertical compression (since it's not as steep as \( \sqrt{x} \)). The domain \( x\geq2 \) and the vertical compression by \( \frac{1}{3} \) match the transformed function \( g(x)=\frac{1}{3}\sqrt{x - 2} \).
• Option B: The graph starts at or near \( x = 0 \), which doesn't match the horizontal shift of 2 units to the right. So this is incorrect.
• Option C: Similar to Option B, the starting point is around \( x = 0 \), not \( x = 2 \). Incorrect.
• Option D: The graph's shape and starting point don't align with the vertical compression and horizontal shift. Incorrect.

Answer:

A