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]:
[tex]\[ y = 23 - \frac{6^2 + 1}{4 \times \frac{1}{2}} \][/tex]
3. Calculate the denominator [tex]\( 4 \times \frac{1}{2} \)[/tex]:
[tex]\[ 4 \times \frac{1}{2} = 2 \][/tex]
4. Substitute this back into the main formula:
[tex]\[ y = 23 - \frac{36 + 1}{2} \][/tex]
5. Simplify inside the fraction:
[tex]\[ y = 23 - \frac{37}{2} \][/tex]
6. Perform the division:
[tex]\[ \frac{37}{2} = 18.5 \][/tex]
7. Subtract this result from 23:
[tex]\[ y = 23 - 18.5 \][/tex]
[tex]\[ y = 4.5 \][/tex]
So, the equation of the directrix is [tex]\( y = 4.5 \)[/tex].
Therefore, the correct choice is:
[tex]\[ y = 4.5 \][/tex]
