The given function to graph is:

$f(x) = -2\sqrt{x - 4}$

Analysis and Key Points:

1. Domain: The expression under the square root, $x - 4$, must be non-negative. Thus, $x - 4 \geq 0$, which implies $x \geq 4$. The domain of the function is $[4, \infty)$.

2. Leftmost Point: The leftmost point is when $x = 4$. Plugging $x = 4$ into $f(x)$:

   $f(4) = -2\sqrt{4 - 4} = -2\sqrt{0} = 0$

   So, the point is $(4, 0)$.

3. Additional Points: Let's choose three additional points in the domain $x \geq 4$.

   • Point 1: Let $x = 5$.

     $f(5) = -2\sqrt{5 - 4} = -2\sqrt{1} = -2$ Point: $(5, -2)$.

   • Point 2: Let $x = 8$.

     $f(8) = -2\sqrt{8 - 4} = -2\sqrt{4} = -4$ Point: $(8, -4)$.

   • Point 3: Let $x = 12$.

     $f(12) = -2\sqrt{12 - 4} = -2\sqrt{8} = -2\sqrt{4 \cdot 2} = -4\sqrt{2} \approx -5.66$ Point: $(12, -5.66)$.

Points to Plot:

• $(4, 0)$
• $(5, -2)$
• $(8, -4)$
• $(12, -5.66)$

You can now use these points to plot the function on the graph.

Would you like more details or have any questions?

Related Questions:

1. What is the range of the function $f(x) = -2\sqrt{x - 4}$?
2. How does the graph of $f(x) = -2\sqrt{x - 4}$ compare to $g(x) = \sqrt{x}$?
3. What are the x-intercepts and y-intercepts of the function?
4. How does changing the coefficient $-2$ affect the shape of the graph?
5. What transformations are applied to the parent function $y = \sqrt{x}$?

Tip: When graphing square root functions, always consider the domain and transformations such as vertical stretching or reflecting.