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 (1)
The axis of symmetry for a quadratic function f ( x ) = a x 2 + b x + c is given by x = − 2 a b .
Substitute the given value a = 3 into the formula: x = − 6 b .
Set the expression equal to the given axis of symmetry: − 6 b = 2 3 .
Solve for b : b = − 9 . The final answer is − 9 .
Explanation
Problem Analysis 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 .
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 c = 4 .
Substituting the Value of a Substituting a = 3 into the axis of symmetry formula, we get x = − 2 ( 3 ) b = − 6 b .
Setting up the Equation We are given that the axis of symmetry is x = 2 3 . Therefore, we can set up the equation − 6 b = 2 3 .
Solving for b To solve for b , we multiply both sides of the equation by − 6 : b = − 6 × 2 3 = − 2 18 = − 9.
Final Answer Therefore, the value of b is − 9 .
Examples
Understanding the axis of symmetry is crucial in various applications, such as optimizing the design of parabolic reflectors in satellite dishes or solar ovens. For instance, if you're building a solar oven, knowing the axis of symmetry helps you position the reflective surface to concentrate sunlight efficiently onto the cooking area. Similarly, in sports, understanding the parabolic trajectory of a ball can help athletes optimize their throwing or hitting techniques. The axis of symmetry provides a key reference point for analyzing and predicting these parabolic paths.
