Question

which of the following describes the graph of $y = \sqrt{-4x - 36}$ compared to the parent square root function? \
\bigcirc stretched by a factor of 2, reflected over the x - axis, and translated 9 units right \
\bigcirc stretched by a factor of 2, reflected over the x - axis, and translated 9 units left \
\bigcirc stretched by a factor of 2, reflected over the y - axis, and translated 9 units right \
\bigcirc stretched by a factor of 2, reflected over the y - axis, and translated 9 units left

Explanation:

Step 1: Simplify the function

First, we simplify the expression inside the square root:
$$y = \sqrt{-4x - 36}=\sqrt{-4(x + 9)}=\sqrt{4(-x - 9)} = 2\sqrt{-x - 9}$$
The parent square root function is \(y=\sqrt{x}\).

Step 2: Analyze the transformations

• Stretch factor: The coefficient \(2\) in front of the square root indicates a vertical stretch by a factor of \(2\).
• Reflection: The negative sign inside the square root (in the simplified form \(\sqrt{-x - 9}\)) indicates a reflection over the \(y\)-axis (since the transformation for reflection over \(y\)-axis is replacing \(x\) with \(-x\)).
• Translation: We have \(-x - 9=-(x + 9)\), so the function is \(2\sqrt{-(x + 9)}\). Comparing to the parent function \(y = \sqrt{x}\), the \(x+9\) inside the square root (after reflection transformation) indicates a translation of \(9\) units to the left (because the transformation \(y=\sqrt{-(x + h)}\) for \(h>0\) is a translation \(h\) units left).

Answer:

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