Question

which of the following describes the graph of $y = \sqrt{-4x - 36}$ compared to the parent square root function?

• stretched by a factor of 2, reflected over the x-axis, and translated 9 units right
• stretched by a factor of 2, reflected over the x-axis, and translated 9 units left
• stretched by a factor of 2, reflected over the y-axis, and translated 9 units right
• stretched by a factor of 2, reflected over the y-axis, and translated 9 units left

Explanation:

Step1: Simplify the function

First, we simplify \( y = \sqrt{-4x - 36} \). Factor out \(-4\) from the radicand: \( y=\sqrt{-4(x + 9)}=\sqrt{4}\cdot\sqrt{- (x + 9)} = 2\sqrt{-(x + 9)} \). The parent square root function is \( y=\sqrt{x} \).

Step2: Analyze transformations

• Vertical stretch: The coefficient \( 2 \) in front of the square root indicates a vertical stretch by a factor of \( 2 \) (since \( a = 2>1 \) in the form \( y=a\sqrt{b(x - h)}+k \)).
• Reflection: The negative sign inside the square root (in front of \( x \)) indicates a reflection over the \( y \)-axis (because the transformation \( \sqrt{-x} \) is a reflection over the \( y \)-axis of \( \sqrt{x} \)).
• Horizontal translation: The \( (x + 9) \) inside the square root (compared to \( x \) in the parent function) indicates a horizontal translation. For the form \( y=\sqrt{-(x - h)} \), here \( h=- 9 \), so the translation is \( 9 \) units to the left (since \( h=-9 \), and the formula for horizontal translation is \( x\to x - h \), so \( x\to x+9 \) means shifting left by \( 9 \) units).

Answer:

stretched by a factor of 2, reflected over the \( y \)-axis, and translated 9 units left (the fourth option: stretched by a factor of 2, reflected over the \( y \)-axis, and translated 9 units left)