Question

quiz: graphing radical functions
hs: algebra 2b m (sequential) / 2 inverse functions

1. rewrite $y = \sqrt{9x - 36} - 4$ to make it easy to graph using a translation. describe the graph.

\bigcirc the equation is $y = \sqrt{x - 4} - 4$. it is the graph of $y = \sqrt{x}$ translated 4 units left and 4 units down.
\bigcirc the equation is $y = 3\sqrt{x - 4} - 4$. it is the graph of $y = 3\sqrt{x}$ translated 4 units right and 4 units down.
\bigcirc the equation is $y = 3\sqrt{x - 4} - 4$. it is the graph of $y = 3\sqrt{x}$ translated 4 units left and 4 units down.
\bigcirc the equation is $y = \sqrt{x - 4} - 4$. it is the graph of $y = \sqrt{x}$ translated 4 units right and 4 units down.

Explanation:

Step1: Simplify the original function

Given \( y=\sqrt{9x - 36}-4 \), factor out 9 from the radicand: \( y=\sqrt{9(x - 4)}-4 \).
Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we have \( \sqrt{9(x - 4)}=\sqrt{9}\cdot\sqrt{x - 4}=3\sqrt{x - 4} \). So the simplified function is \( y = 3\sqrt{x - 4}-4 \).

Step2: Analyze the translation

The parent function is \( y = 3\sqrt{x} \). For a radical function of the form \( y=a\sqrt{x - h}+k \), the graph of \( y=a\sqrt{x} \) is translated \( h \) units to the right (if \( h>0 \)) and \( k \) units down (if \( k<0 \)). Here, \( h = 4 \) and \( k=- 4 \), so the graph of \( y = 3\sqrt{x} \) is translated 4 units right and 4 units down to get \( y = 3\sqrt{x - 4}-4 \).

Answer:

The second option (the one with \( y = 3\sqrt{x - 4}-4 \) and description about translation 4 units right and 4 units down)