Question

graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. start with the graph of the basic function shown below.\\( h(x) = \sqrt{x - 6} \\)\
choose the correct graph below.\
\\( \boldsymbol{\text{o a.}} \\) graph a\
\\( \boldsymbol{\text{o b.}} \\) graph b\
\\( \boldsymbol{\text{o c.}} \\) graph c\
\\( \boldsymbol{\text{o d.}} \\) graph d

Explanation:

Step1: Analyze the parent function

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

Step2: Determine the transformation

For the function \( h(x)=\sqrt{x - 6} \), this is a horizontal shift of the parent function \( y=\sqrt{x} \). The general form for a horizontal shift is \( y = \sqrt{x - h} \), which shifts the graph \( h \) units to the right when \( h>0 \). Here, \( h = 6 \), so we shift the graph of \( y=\sqrt{x} \) 6 units to the right.

Step3: Analyze the domain and key points

The domain of \( h(x)=\sqrt{x - 6} \) is \( x - 6\geq0\) or \( x\geq6 \). The starting point (vertex) of the square - root function after the shift will be at \( (6,0) \), and the function will be increasing (since there is no reflection or vertical stretch/compression that would reverse the direction) as \( x \) increases beyond 6.

Now let's analyze the graphs:

• Graph A: The starting point seems to be around \( x = 6 \) (since the graph starts at \( x\approx6 \) on the x - axis) and it is an increasing function (as \( x \) increases, \( y \) increases), which matches the behavior of \( h(x)=\sqrt{x - 6} \).
• Graph B: The starting point is at a negative \( x \) - value, which does not match the domain \( x\geq6 \).
• Graph C: The function is decreasing, which is not the case for \( h(x)=\sqrt{x - 6} \) (the square - root function \( \sqrt{u} \) with \( u=x - 6 \) and \( u\) increasing as \( x \) increases will have an increasing \( y \) - value).
• Graph D: The starting point is at \( x = 0 \), which does not match the domain \( x\geq6 \).

Answer:

A. The graph labeled A (with the starting point around \( x = 6 \) and increasing as \( x \) increases)