To determine the coordinates of the y-intercept of the graph of the equation [tex]\( y = -\frac{1}{3} x^2 - 2 x + 1 \)[/tex], we need to follow these steps:
1. Know What the Y-Intercept Is: The y-intercept is the point where the graph intersects the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the Equation: We plug [tex]\( x = 0 \)[/tex] into the given equation to find the corresponding [tex]\( y \)[/tex]-value.
[tex]\[
y = -\frac{1}{3} (0)^2 - 2(0) + 1
\][/tex]
3. Simplify the Equation:
[tex]\[
y = -\frac{1}{3} \cdot 0 - 2 \cdot 0 + 1 = 0 - 0 + 1 = 1
\][/tex]
4. Determine the Coordinates: The coordinates of the y-intercept are therefore [tex]\( (0, 1) \)[/tex].
Given the choices:
(A) [tex]\((-6, 1)\)[/tex]
(B) [tex]\((-3, 4)\)[/tex]
(C) [tex]\((0, -1)\)[/tex]
(D) [tex]\((0, 1)\)[/tex]
The correct choice is (D) [tex]\((0, 1)\)[/tex].
Thus, the coordinates of the y-intercept of the graph are [tex]\( \boxed{(0, 1)} \)[/tex].
