The given equation of the graph is [tex]\( y = -\frac{1}{2} x^2 + 2 x + 3 \)[/tex].
To determine which characteristic of the graph is displayed as a constant or coefficient in the equation, we will examine the given options:
### (A) [tex]\( x \)[/tex]-intercept(s)
The [tex]\( x \)[/tex]-intercepts are the points where the graph crosses the [tex]\( x \)[/tex]-axis. These are found by setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]. However, the [tex]\( x \)[/tex]-intercepts are not explicitly displayed as constants or coefficients in the given equation.
### (B) Minimum [tex]\( x \)[/tex]-value
The minimum [tex]\( x \)[/tex]-value typically refers to the vertex's [tex]\( x \)[/tex]-coordinate in a parabolic graph. The [tex]\( x \)[/tex]-coordinate of the vertex is found using [tex]\( x = -\frac{b}{2a} \)[/tex], but this value is calculated rather than being directly displayed as a coefficient or constant in the equation.
### (C) [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept is the point where the graph crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the equation, we have:
[tex]\[ y = -\frac{1}{2} (0)^2 + 2(0) + 3 = 3 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is [tex]\( 3 \)[/tex], and this is directly displayed as a constant in the equation.
### (D) [tex]\( y \)[/tex]-coordinate of the vertex
The [tex]\( y \)[/tex]-coordinate of the vertex can be determined by substituting the [tex]\( x \)[/tex]-coordinate of the vertex back into the equation. However, this value requires a calculation and is not directly given as a constant or coefficient in the equation.
Based on the analysis, the [tex]\( y \)[/tex]-intercept, which is [tex]\( 3 \)[/tex], is the characteristic of the graph that is displayed as a constant in the equation.
Therefore, the correct answer is:
(c) [tex]\( y \)[/tex]-intercept
