What is the axis of symmetry of the function [tex]$f(x)=-(x+9)(x-21)$[/tex]?
A. [tex]$x=-6$[/tex]
B. [tex]$x=6$[/tex]
C. [tex]$x=-15$[/tex]
D. [tex]$x=15$[/tex]
Answer (1)
Identify the roots of the quadratic function: x = − 9 and x = 21 .
Calculate the axis of symmetry by averaging the roots: x = 2 − 9 + 21 .
Simplify the expression: x = 2 12 = 6 .
The axis of symmetry is x = 6 .
Explanation
Understanding the Problem We are given the function f ( x ) = − ( x + 9 ) ( x − 21 ) and asked to find its axis of symmetry. The axis of symmetry is the vertical line that passes through the vertex of the parabola. Since the function is given in factored form, we can easily find the roots of the function, which are the x -intercepts. These roots are x = − 9 and x = 21 . The axis of symmetry is located exactly in the middle of the two roots.
Calculating the Axis of Symmetry To find the axis of symmetry, we take the average of the two roots: x = 2 − 9 + 21 x = 2 12 x = 6 Thus, the axis of symmetry is x = 6 .
Final Answer The axis of symmetry of the function f ( x ) = − ( x + 9 ) ( x − 21 ) is x = 6 .
Examples
Imagine you're designing a symmetric archway. The function given represents the shape of the arch, and finding the axis of symmetry helps you determine the center point of the arch. This ensures that both sides of the arch are balanced and aesthetically pleasing. Understanding the axis of symmetry is crucial in architecture and design for creating balanced and stable structures.
