Question

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

1. rewrite $y = \sqrt{25x - 125} + 5$ to make it easy to graph using a translation. describe the graph.

• the equation is $y = 5\sqrt{x - 5} + 5$. it is the graph of $y = 5\sqrt{x}$ translated 5 units left and 5 units up.
• the equation is $y = \sqrt{x - 5} + 5$. it is the graph of $y = \sqrt{x}$ translated 5 units left and 5 units up.
• the equation is $y = \sqrt{x + 5} + 5$. it is the graph of $y = \sqrt{x}$ translated 5 units right and 5 units up.
• the equation is $y = 5\sqrt{x - 5} + 5$. it is the graph of $y = 5\sqrt{x}$ translated 5 units right and 5 units up.

Explanation:

Step1: Simplify the original function

Start with the original function \( y=\sqrt{25x - 125}+5 \). Factor out 25 from the radicand: \( \sqrt{25(x - 5)}+5 \). Since \( \sqrt{25(x - 5)}=\sqrt{25}\cdot\sqrt{x - 5}=5\sqrt{x - 5} \), the function simplifies to \( y = 5\sqrt{x - 5}+5 \).

Step2: Analyze the translation

The parent function here is \( y = 5\sqrt{x} \). For a radical function of the form \( y=a\sqrt{x - h}+k \), the graph is a translation of \( y=a\sqrt{x} \). Here, \( h = 5 \) and \( k = 5 \). A positive \( h \) means a translation \( h \) units to the right, and a positive \( k \) means a translation \( k \) units up. So \( y = 5\sqrt{x - 5}+5 \) is \( y = 5\sqrt{x} \) translated 5 units right and 5 units up.

Answer:

The fourth option (the last one): The equation is \( y = 5\sqrt{x - 5}+5 \), It is the graph of \( y = 5\sqrt{x} \) translated 5 units right and 5 units up.