To determine the focal width of the given parabola, we start by identifying the equation and its components.
The given equation is:
[tex]\[ y = (x + 2)^2 - 3 \][/tex]
This equation is already in the vertex form of a parabola:
[tex]\[ y = a(x - h)^2 + k \][/tex]
From the equation [tex]\( y = (x + 2)^2 - 3 \)[/tex], we can extract the values of [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( h = -2 \)[/tex] (since [tex]\( x + 2 = x - (-2) \)[/tex])
- [tex]\( k = -3 \)[/tex]
We need to find the focal width of the parabola. The focal width (also called the latus rectum) of a parabola given by the equation [tex]\( y = a(x - h)^2 + k \)[/tex] is determined by the parameter [tex]\( a \)[/tex]. Specifically, the focal width is given by the absolute value of [tex]\( 4a \)[/tex].
Given:
[tex]\[ a = 1 \][/tex]
Then the focal width [tex]\( |4a| \)[/tex] is calculated as:
[tex]\[ |4a| = |4 \cdot 1| = 4 \][/tex]
Therefore, the length of the focal width of the parabola is:
[tex]\[ 4 \text{ units} \][/tex]
So, the correct answer is:
[tex]\[ 4 \text{ units} \][/tex]
