To find the value of [tex]\( b \)[/tex] in the standard form of the quadratic function [tex]\( f(t) = 3(2t - 1)^2 + 4 \)[/tex], we follow a systematic approach to rewrite the function and identify its coefficients.
First, let's take the given expression and start by expanding it:
1. Expand [tex]\( (2t - 1)^2 \)[/tex]:
[tex]\[ (2t - 1)^2 = (2t - 1)(2t - 1) = 4t^2 - 4t + 1 \][/tex]
2. Multiply this result by 3:
[tex]\[ 3 \cdot (4t^2 - 4t + 1) = 12t^2 - 12t + 3 \][/tex]
3. Add the constant term (4):
[tex]\[ 12t^2 - 12t + 3 + 4 = 12t^2 - 12t + 7 \][/tex]
Now, the function is in standard quadratic form [tex]\( f(t) = at^2 + bt + c \)[/tex], where:
- [tex]\( a = 12 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Thus, the value of [tex]\( b \)[/tex] in the standard form is [tex]\( -12 \)[/tex].
