To determine why the graph of the equation [tex]\( g(x) = -(x+1)^2 - 3 \)[/tex] is a parabola opening downward, let's start by examining the structure of the equation.
1. Identify the Standard Form:
The equation given is in the form [tex]\( g(x) = -(x+1)^2 - 3 \)[/tex]. This resembles the standard form for a parabola, which is generally written as [tex]\( g(x) = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
2. Identify the Value of [tex]\( a \)[/tex]:
In the equation [tex]\( g(x) = -(x+1)^2 - 3 \)[/tex], we can see that the coefficient of the squared term [tex]\( (x+1)^2 \)[/tex] is [tex]\( -1 \)[/tex]. This coefficient is denoted by [tex]\( a \)[/tex].
Therefore, [tex]\( a = -1 \)[/tex].
3. Determine the Direction of the Parabola:
The direction in which a parabola opens is determined by the sign of [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Since [tex]\( a = -1 \)[/tex] in this case, and [tex]\( -1 < 0 \)[/tex], the parabola opens downward.
So, to summarize:
The parabola opens downward since [tex]\( a = -1 \)[/tex] and this value is less than 0.
