Sure, I'll provide a step-by-step solution to determine the value of [tex]\( b \)[/tex] for the given quadratic function and its axis of symmetry.
Given the quadratic function:
[tex]\[ f(x) = \frac{1}{4} x^2 + bx + 10 \][/tex]
The formula for the axis of symmetry of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In this function, the coefficients are:
[tex]\[ a = \frac{1}{4} \][/tex]
[tex]\[ b = b \][/tex] (We need to determine this)
[tex]\[ c = 10 \][/tex]
We are given that the axis of symmetry is:
[tex]\[ x = 6 \][/tex]
Using the formula for the axis of symmetry:
[tex]\[ 6 = -\frac{b}{2 \cdot \frac{1}{4}} \][/tex]
First, calculate the denominator:
[tex]\[ 2 \cdot \frac{1}{4} = \frac{1}{2} \][/tex]
Now substitute this value back into the equation:
[tex]\[ 6 = -\frac{b}{\frac{1}{2}} \][/tex]
To isolate [tex]\( b \)[/tex], multiply both sides by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ 6 \times \frac{1}{2} = -b \][/tex]
[tex]\[ 3 = -b \][/tex]
Multiply both sides by [tex]\( -1 \)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[ b = -3 \][/tex]
Therefore, the value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{-3} \][/tex]
