What is the axis of symmetry of the function [tex]f(x)=-(x+9)(x-21)[/tex]?
A. [tex]x=-15[/tex]
B. [tex]x=-6[/tex]
C. [tex]x=6[/tex]
D. [tex]x=15[/tex]
Answer (2)
Identify the roots of the function f ( x ) = − ( x + 9 ) ( x − 21 ) as x = − 9 and x = 21 .
Calculate the midpoint of the roots using the formula 2 x 1 + x 2 .
Substitute the roots into the formula: 2 − 9 + 21 = 2 12 = 6 .
The axis of symmetry is x = 6 , so the final answer is 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. The axis of symmetry will be exactly in the middle of the two roots.
Finding the Roots The roots of the function are the values of x that make f ( x ) = 0 . Thus, we have − ( x + 9 ) ( x − 21 ) = 0 . This equation is satisfied when x + 9 = 0 or x − 21 = 0 . Solving these equations, we find the roots are x = − 9 and x = 21 .
Calculating the Axis of Symmetry The axis of symmetry is the vertical line that passes through the midpoint of the roots. The midpoint of the roots x 1 and x 2 is given by 2 x 1 + x 2 . In our case, the roots are − 9 and 21 , so the midpoint is 2 − 9 + 21 = 2 12 = 6.
Final Answer Therefore, the axis of symmetry is the vertical line x = 6 .
Examples
Understanding the axis of symmetry is crucial in various real-world applications. For instance, when designing suspension bridges, engineers use parabolic curves to distribute weight evenly. The axis of symmetry helps determine the bridge's center point, ensuring balanced support and stability. Similarly, in architecture, knowing the axis of symmetry aids in creating symmetrical and aesthetically pleasing structures, like arches or domes, where balance and visual harmony are essential. In physics, projectile motion follows a parabolic path, and the axis of symmetry helps determine the maximum height and range of the projectile, useful in fields like sports and ballistics.
The axis of symmetry for the function f ( x ) = − ( x + 9 ) ( x − 21 ) is found by calculating the midpoint of the roots x = − 9 and x = 21 . This calculation yields x = 6 as the axis of symmetry. Thus, the chosen multiple-choice option is C: x = 6 .
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