Question

rewrite the function to make it easy to graph using transformations of its parent function. describe the graph.

$y = \sqrt{25x - 50} - 3$

$y = 5\sqrt{x - 2} - 3$ (simplify your answer.)

the graph \\(\boldsymbol{\bigtriangledown}\\) vertically by a factor of \\(\boldsymbol{\square}\\) and is translated \\(\boldsymbol{\square}\\) unit(s) to the \\(\boldsymbol{\bigtriangledown}\\) and \\(\boldsymbol{\square}\\) unit(s) \\(\boldsymbol{\bigtriangledown}\\)

Explanation:

Step1: Identify Parent Function

The parent function for square root functions is \( y = \sqrt{x} \). The given function is \( y = 5\sqrt{x - 2} - 3 \).

Step2: Analyze Vertical Stretch/Compression

The coefficient \( 5 \) in front of \( \sqrt{x - 2} \) indicates a vertical stretch. For a function \( y = a\sqrt{x} \), if \( |a|>1 \), it's a vertical stretch by factor \( |a| \). So here, vertical stretch by factor \( 5 \).

Step3: Analyze Horizontal Translation

The \( x - 2 \) inside the square root: for \( y=\sqrt{x - h} \), the graph shifts \( h \) units to the right (if \( h>0 \)). Here \( h = 2 \), so translation \( 2 \) units to the right.

Step4: Analyze Vertical Translation

The \( - 3 \) at the end: for \( y=\sqrt{x}+k \), if \( k<0 \), it shifts \( |k| \) units down. Here \( k=-3 \), so translation \( 3 \) units down.

Answer:

The graph \(\boldsymbol{\text{is stretched}}\) vertically by a factor of \(\boldsymbol{5}\) and is translated \(\boldsymbol{2}\) unit(s) to the \(\boldsymbol{\text{right}}\) and \(\boldsymbol{3}\) unit(s) \(\boldsymbol{\text{down}}\).