Question

use transformations to graph the function.
$l(x)=-sqrt{-x}-6$

Explanation:

Step1: Identify parent function

Parent function: $f(x)=\sqrt{x}$

Step2: Reflect over y-axis

Transform to $f_1(x)=\sqrt{-x}$

Step3: Reflect over x-axis

Transform to $f_2(x)=-\sqrt{-x}$

Step4: Shift down 6 units

Transform to $L(x)=-\sqrt{-x}-6$

Step5: Plot key points

• For parent $f(x)=\sqrt{x}$: $(0,0), (1,1), (4,2), (9,3)$
• After y-reflect: $(0,0), (-1,1), (-4,2), (-9,3)$
• After x-reflect: $(0,0), (-1,-1), (-4,-2), (-9,-3)$
• After down shift: $(0,-6), (-1,-7), (-4,-8), (-9,-9)$

Answer:

The graph of $L(x)=-\sqrt{-x}-6$ is obtained by:

1. Reflecting $y=\sqrt{x}$ over the y-axis,
2. Reflecting the result over the x-axis,
3. Shifting the final graph down 6 units.

Key points to plot: $(0,-6), (-1,-7), (-4,-8), (-9,-9)$, with the curve defined for $x\leq0$, decreasing as $x$ decreases, approaching negative infinity as $x\to-\infty$.