First, recall the general form of a parabola equation: [tex]\( y = ax^2 + bx + c \)[/tex]. For the given parabola equation [tex]\( y = \frac{1}{2} x^2 + 6 x + 23 \)[/tex], we can identify the coefficients: - [tex]\( a = \frac{1}{2} \)[/tex] - [tex]\( b = 6 \)[/tex] - [tex]\( c = 23 \)[/tex]
The formula to find the directrix of a parabola in the format [tex]\( y = ax^2 + bx + c \)[/tex] is:
[tex]\[ y = c - \frac{b^2 + 1}{4a} \][/tex]
Let's break it down step by step:
1. Begin by substituting the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula for the directrix:
[tex]\[ y = c - \frac{b^2 + 1}{4a} \][/tex]
2. Substitute [tex]\( a = \frac{1}{2} \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = 23 \)[/tex]: