To find the value of [tex]\( b \)[/tex] in the quadratic function [tex]\( f(x) = \frac{1}{4}x^2 + bx + 10 \)[/tex] given that the axis of symmetry is [tex]\( x = 6 \)[/tex], we can follow these steps:
1. Recall the formula for the axis of symmetry:
For a quadratic function in standard form [tex]\( ax^2 + bx + c \)[/tex], the axis of symmetry is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
2. Identify the coefficients:
In the given function [tex]\( f(x) = \frac{1}{4}x^2 + bx + 10 \)[/tex]:
- [tex]\( a = \frac{1}{4} \)[/tex]
- [tex]\( b = ? \)[/tex] (this is what we need to find)
- [tex]\( c = 10 \)[/tex]
3. Use the given axis of symmetry:
We are given that the axis of symmetry is [tex]\( x = 6 \)[/tex]. Therefore, we can substitute this into the formula for the axis of symmetry:
[tex]\[
6 = -\frac{b}{2 \cdot \frac{1}{4}}
\][/tex]
4. Simplify the equation:
First, multiply the denominator [tex]\( 2 \cdot \frac{1}{4} \)[/tex]:
[tex]\[
2 \cdot \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\][/tex]
Now the equation becomes:
[tex]\[
6 = -\frac{b}{\frac{1}{2}}
\][/tex]
5. Solve for [tex]\( b \)[/tex]:
To isolate [tex]\( b \)[/tex], first invert the division by [tex]\( \frac{1}{2} \)[/tex] (which is equivalent to multiplying by 2):
[tex]\[
6 = -b \cdot 2
\][/tex]
[tex]\[
6 = -2b
\][/tex]
Divide both sides by -2:
[tex]\[
b = \frac{6}{-2}
\][/tex]
[tex]\[
b = -3
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] is [tex]\( -3 \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{-3}
\][/tex]
