Question

guided practice
graphing radical functions
rewrite $y = \sqrt{4x + 20} + 5$ to make it easy to identify the transformations and describe them.
a. rewritten, $y = \sqrt{x - 5} + 5$. it is the graph of $y = \sqrt{x}$ translated 5 units left and 5 units up.
b. rewritten, $y = 2\sqrt{x + 5} + 5$. it is the graph of $y = 2\sqrt{x}$ translated 5 units right and 5 units up.
c. rewritten, $y = 2\sqrt{x + 5} + 5$. it is the graph of $y = 2\sqrt{x}$ translated 5 units left and 5 units up.
d. rewritten, $y = \sqrt{x + 5} + 5$. it is the graph of $y = \sqrt{x}$ translated 5 units right and 5 units up.

Explanation:

1. First, simplify \( y=\sqrt{4x + 20}+5 \). Factor out 4 from the radicand: \( \sqrt{4(x + 5)}+5 \). Since \( \sqrt{4}=2 \), this becomes \( 2\sqrt{x + 5}+5 \).
2. For the parent function \( y = 2\sqrt{x} \), the transformation \( y=2\sqrt{x + h}+k \) means a horizontal translation of \( h \) units (left if \( h>0 \)) and vertical translation of \( k \) units (up if \( k>0 \)). Here, \( h = 5 \) (so 5 units left) and \( k = 5 \) (so 5 units up). Options A, B, D have errors in simplification or transformation description.

Answer:

C. Rewritten, \( y = 2\sqrt{x + 5}+5 \). It is the graph of \( y = 2\sqrt{x} \) translated 5 units left and 5 units up.