The axis of symmetry for the graph of the function [tex]$f(x)=3 x^2+b x+4$[/tex] is [tex]$x=\frac{3}{2}$[/tex]. What is the value of [tex]$b$[/tex]?
A. -18
B. -9
C. 9
D. 18
Answer (2)
The axis of symmetry for a quadratic function f ( x ) = a x 2 + b x + c is given by x = − 2 a b .
Given f ( x ) = 3 x 2 + b x + 4 and the axis of symmetry x = 2 3 , we set up the equation 2 3 = − 2 ( 3 ) b .
Solving for b , we get b = − 6 × 2 3 .
Therefore, the value of b is − 9 .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 3 x 2 + b x + 4 and its axis of symmetry x = 2 3 . Our goal is to find the value of b .
Recalling the Axis of Symmetry Formula The axis of symmetry for a quadratic function in the form f ( x ) = a x 2 + b x + c is given by the formula x = − 2 a b . In our case, a = 3 and the axis of symmetry is x = 2 3 .
Setting up the Equation We can set up the equation 2 3 = − 2 ( 3 ) b , which simplifies to 2 3 = − 6 b .
Solving for b To solve for b , we can multiply both sides of the equation by − 6 : − 6 \t 2 3 = − 6 \t − 6 b This simplifies to b = − 9 .
Final Answer Therefore, the value of b is − 9 .
Examples
Understanding the axis of symmetry is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, knowing the axis of symmetry helps optimize the focus of light or signal. Similarly, in architecture, the axis of symmetry can guide the construction of arches and suspension bridges, ensuring structural balance and aesthetic appeal. By determining the value of 'b' in a quadratic function, we can accurately predict and control the symmetry of these designs, leading to more efficient and visually pleasing outcomes.
The value of b in the quadratic function f ( x ) = 3 x 2 + b x + 4 is − 9 . This is derived from the axis of symmetry formula x = − 2 a b and substituting the known values. Thus, the correct option is B. -9.
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