Question

y = -2√(x - 5) what are the transformations of this functions compared to the parent function? options: translated right 5, translated down 2; reflected over the x-axis, vertical stretch 2, and translated left 5; reflected over the x-axis, vertical stretch 2, and translated right 5; translated right 5, translated down 2, and reflected over the x-axis

Explanation:

The parent function of a square root function is \( y = \sqrt{x} \). Let's analyze the transformations step by step for the given function \( y = -2\sqrt{x - 5} \):

Step 1: Horizontal Translation

For a function of the form \( y = \sqrt{x - h} \), if \( h>0 \), the graph is translated \( h \) units to the right. In our function, \( h = 5 \), so the graph is translated 5 units to the right.

Step 2: Vertical Stretch/Reflection

The coefficient \( -2 \) in front of the square root. The negative sign indicates a reflection over the \( x \)-axis. The absolute value of the coefficient, \( | - 2| = 2 \), indicates a vertical stretch by a factor of 2.

Now let's check the options:

• Option 1: Says translated down 2, but there is no vertical translation (the function is \( y=-2\sqrt{x - 5} \), no constant term added or subtracted at the end), so this is incorrect.
• Option 2: Says translated left 5, but we have a translation right 5, so this is incorrect.
• Option 3: Says reflected over the \( x \)-axis (because of the negative sign), vertical stretch 2 (because of \( | - 2|=2 \)) and translated right 5 (because of \( x - 5 \)), which matches our analysis.
• Option 4: Says translated down 2, but there is no vertical translation, so this is incorrect.

Answer:

C. Reflected over the x - axis, vertical stretch 2, and translated right 5 (assuming the orange option is option C, if the options are labeled as A: Green, B: Purple, C: Orange, D: Teal, then the answer is C. Reflected over the x - axis, vertical stretch 2, and translated right 5)