Question

y = -2\sqrt{x - 5} what are the transformations of this function compared to the parent function? translated right 5, translated down 2; reflected over the x - axis, vertical stretch 2, and translated right 5; translated right 5, translated down 2, and reflected over the x - axis; reflected over the x - axis, vertical stretch 2, and translated left 5

Explanation:

Step1: Recall parent function

The parent function for square root functions is \( y = \sqrt{x} \).

Step2: Analyze transformations

• Reflection: The negative sign in front (\(-2\)) indicates a reflection over the \(x\)-axis.
• Vertical Stretch: The coefficient \(2\) (absolute value) indicates a vertical stretch by a factor of \(2\).
• Horizontal Translation: The \(x - 5\) inside the square root indicates a translation to the right by \(5\) units (since for \(y=\sqrt{x - h}\), it's a shift right by \(h\) when \(h>0\)).

Now let's check the options:

• Option 1: No reflection or stretch, incorrect.
• Option 2: Matches reflection over \(x\)-axis, vertical stretch by 2, and translation right 5.
• Option 3: No vertical stretch factor 2 (the -2 is reflection and stretch, but the "translated down 2" is wrong as there's no vertical shift term), incorrect.
• Option 4: Translation left 5 is wrong (it's \(x - 5\) so right 5), incorrect.

Answer:

B. Reflected over the x - axis, vertical stretch 2, and translated right 5 (assuming the purple option is B, adjust identifier if needed based on actual option labels, but the description is "Reflected over the x - axis, vertical stretch 2, and translated right 5")