Which direction does the graph of the parabola described by the function [tex]g(x)=-\frac{1}{2} x^2+x+0.5[/tex] open towards?
A. Up
B. Right
C. Down
D. Left
Answer (2)
Identify the coefficient 'a' of the x 2 term in the quadratic function.
Determine the sign of 'a'.
If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
Since a = − 2 1 is negative, the parabola opens Do w n .
Explanation
Understanding the Problem The given function is g ( x ) = − 2 1 x 2 + x + 0.5 , which represents a parabola. We need to determine the direction in which the parabola opens.
General Form of a Parabola The general form of a parabola is a x 2 + b x + c . The sign of the coefficient 'a' determines the direction of opening. If 0"> a > 0 , the parabola opens upwards, and if a < 0 , the parabola opens downwards.
Identifying the Coefficient In the given function, g ( x ) = − 2 1 x 2 + x + 0.5 , the coefficient of the x 2 term is a = − 2 1 .
Determining the Direction Since a = − 2 1 is negative ( a < 0 ), the parabola opens downwards.
Conclusion Therefore, the graph of the parabola described by the function g ( x ) = − 2 1 x 2 + x + 0.5 opens downwards.
Examples
Understanding the direction a parabola opens is crucial in various real-world applications. For instance, engineers designing suspension bridges use parabolic curves to distribute weight evenly. Knowing whether the parabola opens upwards or downwards helps them determine the optimal cable arrangement to ensure the bridge's stability. Similarly, in projectile motion, the path of an object thrown into the air follows a parabolic trajectory, and understanding the direction of the parabola helps predict the object's landing point.
The graph of the parabola described by the function g ( x ) = − 2 1 x 2 + x + 0.5 opens downwards because the coefficient 'a' of the x 2 term is negative. Therefore, the answer is C. Down .
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