The axis of symmetry for the graph of the function [tex]f(x)=\frac{1}{4} x^2+b x+10[/tex] is [tex]x=6[/tex]. What is the value of [tex]b[/tex]?
Answer (1)
Recall the axis of symmetry formula: x = − 2 a b .
Substitute the given values: 6 = − 2 ( 4 1 ) b .
Simplify the equation: 6 = − 2 b .
Solve for b : b = − 3 .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 4 1 x 2 + b x + 10 and its axis of symmetry x = 6 . 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 = 4 1 and the coefficient of the x term is b (which is what we want to find).
Applying the Formula Plugging in the value of a into the axis of symmetry formula, we get: x = − 2 ( 4 1 ) b = − 2 1 b = − 2 b
Setting up the Equation We are given that the axis of symmetry is x = 6 . Therefore, we can set up the equation: − 2 b = 6
Solving for b Now, we solve for b :
b = − 2 6 = − 3
Final Answer Therefore, the value of b is − 3 .
Examples
Understanding the axis of symmetry is crucial in various applications, such as optimizing the design of parabolic reflectors used in satellite dishes or solar ovens. For instance, if you're designing a solar oven, knowing the axis of symmetry helps you position the heating element to maximize the concentration of sunlight at the focal point, thereby increasing the oven's efficiency. Similarly, in architecture, the axis of symmetry can guide the placement of structural elements to ensure balanced weight distribution and aesthetic appeal. Let's say you want to build a parabolic bridge. The equation of the parabola is y = 4 1 x 2 − 3 x + 10 . The lowest point of the bridge is at x = 6 , and the height at that point is y = 4 1 ( 6 ) 2 − 3 ( 6 ) + 10 = 9 − 18 + 10 = 1 .
