Question

describe the transformations on the following radical function: f(x) = - (x + 1) ^ (1 / 2) - 5 (ignore score)

Explanation:

Step1: Recall Parent Function

The parent radical function is \( y = \sqrt{x} \) (which is \( y = x^{\frac{1}{2}} \)).

Step2: Analyze Horizontal Shift

For the function \( f(x)=-\sqrt{x + 1}-5 \), compare with \( y=\sqrt{x} \). The \( x+1 \) inside the square root means a horizontal shift. The rule for horizontal shift is: if we have \( y=\sqrt{x - h} \), it shifts \( h \) units to the right, and \( y=\sqrt{x+h}=\sqrt{x-(-h)} \) shifts \( h \) units to the left. Here \( h = 1 \), so the graph of \( y = \sqrt{x} \) is shifted 1 unit to the left.

Step3: Analyze Reflection

The negative sign in front of the square root, \( - \sqrt{x + 1} \), means a reflection over the \( x \)-axis. The rule for reflection over the \( x \)-axis is multiplying the function by - 1, so \( y=-f(x) \) reflects \( y = f(x) \) over the \( x \)-axis.

Step4: Analyze Vertical Shift

The \( - 5 \) at the end, \( -\sqrt{x + 1}-5 \), means a vertical shift. The rule for vertical shift is: if we have \( y = f(x)+k \), when \( k<0 \), it shifts \( |k| \) units down. Here \( k=-5 \), so the graph is shifted 5 units down.

Answer:

The parent function \( y = \sqrt{x} \) undergoes three transformations: a horizontal shift 1 unit to the left, a reflection over the \( x \)-axis, and a vertical shift 5 units down.