Question

guided practice
rewrite $y = \sqrt{9x - 27} - 5$ to make it easy to identify the transformations and describe them.
a. rewritten, $y = 3\sqrt{x - 3} - 5$. it is the graph of $y = \sqrt{x}$ translated 3 units left and 5 units down.
b. rewritten, $y = 3\sqrt{x - 3} - 5$. it is the graph of $y = 3\sqrt{x}$ translated 3 units right and 5 units down.
c. rewritten, $y = 3\sqrt{x + 3} - 5$. it is the graph of $y = \sqrt{x}$ translated 3 units right and 5 units down.
d. rewritten, $y = \sqrt{x - 3} - 5$. it is the graph of $y = \sqrt{x}$ translated 3 units left and 5 units down.

Explanation:

1. First, simplify \( y=\sqrt{9x - 27}-5 \). Factor out 9 from the radicand: \( y=\sqrt{9(x - 3)}-5 \).
2. Use the property of square roots \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)): \( \sqrt{9(x - 3)}=\sqrt{9}\cdot\sqrt{x - 3}=3\sqrt{x - 3} \). So the equation becomes \( y = 3\sqrt{x - 3}-5 \).
3. For function transformations:

• The form \( y = f(x - h)+k \) represents a horizontal shift (right if \( h>0 \), left if \( h<0 \)) and vertical shift (up if \( k>0 \), down if \( k<0 \)) of the function \( y = f(x) \).
• Here, comparing \( y = 3\sqrt{x - 3}-5 \) with \( y = 3\sqrt{x}+(- 5) \), we have \( h = 3 \) (so 3 units right) and \( k=-5 \) (so 5 units down) from the parent function \( y = 3\sqrt{x} \).
• Option A is wrong about the direction of horizontal shift and the parent function. Option C has an incorrect radicand (\( x + 3 \) instead of \( x - 3 \)). Option D has an incorrect simplification (missing the factor of 3 from the square root simplification) and wrong shift direction.

Answer:

B. Rewritten, \( y = 3\sqrt{x - 3}-5 \). It is the graph of \( y = 3\sqrt{x} \) translated 3 units right and 5 units down.