To determine the focal width of the parabola given by the equation [tex]\( y = (x+2)^2 - 3 \)[/tex], let's analyze its form and properties step-by-step.
1. Rewrite the given equation in the standard form of a parabola:
The standard form of a parabola with a vertical axis of symmetry is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
In our given equation [tex]\( y = (x+2)^2 - 3 \)[/tex]:
[tex]\[ y = (x - (-2))^2 - 3 \][/tex]
By comparison, we identify:
[tex]\[ h = -2 \][/tex]
[tex]\[ k = -3 \][/tex]
2. Identify the vertex and the direction of the parabola:
From the equation, the vertex of the parabola is at [tex]\( (h, k) = (-2, -3) \)[/tex]. Since the squared term is [tex]\((x + 2)^2\)[/tex], which is always non-negative, and since the leading coefficient (the coefficient of the [tex]\( (x - h)^2 \)[/tex] term) is positive, the parabola opens upwards.
3. Determine the coefficient [tex]\( a \)[/tex]:
The given equation is already in the form [tex]\( y = 1(x + 2)^2 - 3 \)[/tex]:
[tex]\[ a = 1 \][/tex]
4. Relate the coefficient [tex]\( a \)[/tex] to the focal distance [tex]\( p \)[/tex]:
For a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex], the coefficient [tex]\( a \)[/tex] is related to the focal distance [tex]\( p \)[/tex] by the formula:
[tex]\[ a = \frac{1}{4p} \][/tex]
Substituting [tex]\( a = 1 \)[/tex]:
[tex]\[ 1 = \frac{1}{4p} \][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[ 4p = 1 \][/tex]
[tex]\[ p = \frac{1}{4} \][/tex]
5. Calculate the focal width:
The focal width of a parabola is given by [tex]\( 4p \)[/tex]. Substituting our value of [tex]\( p \)[/tex]:
[tex]\[ \text{Focal width} = 4p = 4 \times \frac{1}{4} = 1 \][/tex]
Therefore, the length of the focal width of the parabola is 1 unit.
