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The axis of symmetry for the graph of the function [tex][tex]$f(x) = 3x^2 + bx + 4$[/tex][/tex] is [tex][tex]$x = \frac{3}{2}$[/tex][/tex]. What is the value of [tex]b[/tex]?

A. [tex]-18[/tex]
B. [tex]-9[/tex]
C. [tex]9[/tex]
D. [tex]18[/tex]


Sagot :

To determine the value of [tex]\( b \)[/tex] in the quadratic function [tex]\( f(x) = 3x^2 + bx + 4 \)[/tex], we need to use the fact that the axis of symmetry of a parabola described by the equation [tex]\( ax^2 + bx + c \)[/tex] is given by the formula:

[tex]\[ x = -\frac{b}{2a} \][/tex]

In our given function [tex]\( f(x) = 3x^2 + bx + 4 \)[/tex], the coefficient [tex]\( a \)[/tex] is 3. According to the problem, the axis of symmetry is [tex]\( x = \frac{3}{2} \)[/tex]. Let's set up the equation using the axis of symmetry formula:

[tex]\[ \frac{3}{2} = -\frac{b}{2 \cdot 3} \][/tex]

Simplify the denominator on the right hand side:

[tex]\[ \frac{3}{2} = -\frac{b}{6} \][/tex]

To solve for [tex]\( b \)[/tex], we multiply both sides of the equation by 6:

[tex]\[ 6 \cdot \frac{3}{2} = -b \][/tex]

[tex]\[ 3 \cdot 3 = -b \][/tex]

[tex]\[ 9 = -b \][/tex]

So, multiplying both sides by -1 gives:

[tex]\[ b = -9 \][/tex]

Hence, the value of [tex]\( b \)[/tex] is [tex]\( -9 \)[/tex].

The correct answer is:
[tex]\[ \boxed{-9} \][/tex]