Questions: 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

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

Transcript text: Which of the following describes the graph of $y=\sqrt{-4 x-36}$ 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

Solution

Solution Steps

To analyze the transformation of the function $ y = \sqrt{-4x - 36} $ compared to the parent function $ y = \sqrt{x} $, we need to consider the effects of the coefficients and constants inside the square root. The negative sign in front of the $ 4x $ indicates a reflection over the $ y $-axis. The factor of 4 suggests a horizontal compression by a factor of 2 (since the transformation is inside the square root, it affects the $ x $-values inversely). The constant $-36$ inside the square root indicates a horizontal translation. Solving $-4x - 36 = 0$ gives the translation point.

Step 1: Identify the Function

The given function is $ y = \sqrt{-4x - 36} $. We will analyze how this function transforms from the parent function $ y = \sqrt{x} $.

Step 2: Determine the Reflection

The negative sign in front of $ 4x $ indicates a reflection over the $ y $-axis. Thus, the graph of the function is reflected across the $ y $-axis.

Step 3: Analyze the Horizontal Compression

The coefficient $ -4 $ suggests a horizontal compression. Specifically, since the transformation is inside the square root, the function is compressed by a factor of $ \frac{1}{2} $ (because $ \sqrt{4} = 2 $).

Step 4: Find the Translation

To find the translation, we set the expression inside the square root to zero: \[ -4x - 36 = 0 \implies -4x = 36 \implies x = -9 \] This indicates a horizontal translation of $ 9 $ units to the left.

Final Answer

The transformations of the function $ y = \sqrt{-4x - 36} $ compared to the parent square root function are: reflected over the $ y $-axis, stretched by a factor of $ \frac{1}{2} $, and translated $ 9 $ units left.

Thus, the answer is: $\boxed{\text{stretched by a factor of 2, reflected over the } y\text{-axis, and translated 9 units left}}$

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