Question

f(x) = -√(x + 1) + 2
describe the transformation.
options:
right 1 unit, reflect over x - axis, up 2 units
left 1 unit, reflect over x - axis, up 2 units
right 1 unit, reflect over y - axis, up 2 units
left 1 unit, reflect over y - axis, down 2 units

Explanation:

Step1: Recall parent function

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

Step2: Analyze horizontal shift

For a function \( y=\sqrt{x + h} \), if \( h>0 \), it shifts left \( h \) units. Here, we have \( \sqrt{x + 1} \), so \( h = 1>0 \), which means a shift left 1 unit from the parent function \( y=\sqrt{x} \).

Step3: Analyze reflection

The negative sign in front of the square root, \( -\sqrt{x + 1} \), indicates a reflection over the \( x \)-axis (since reflecting \( y = f(x) \) over \( x \)-axis gives \( y=-f(x) \)).

Step4: Analyze vertical shift

The \( + 2 \) at the end, \( -\sqrt{x + 1}+2 \), indicates a vertical shift up 2 units (since adding a constant \( k \) to \( y = f(x) \) gives \( y=f(x)+k \), and if \( k>0 \), it's a shift up \( k \) units).

Answer:

B. Left 1 unit, Reflect over x - axis, Up 2 units