Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on its graph, depending on the direction it opens. For the quadratic equation in standard form, y = ax^2 + bx + c, the vertex can be found using the formula x = -b/(2a). In this case, the vertex helps determine the maximum or minimum value of the function, which is crucial for identifying the range.
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Direction of Opening
The direction in which a parabola opens is determined by the coefficient 'a' in the quadratic equation. If 'a' is positive, the parabola opens upwards, indicating that the vertex is the minimum point. Conversely, if 'a' is negative, as in this case, the parabola opens downwards, meaning the vertex is the maximum point, which affects the range of the function.
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Domain and Range
The domain of a function refers to all possible input values (x-values) that can be used, while the range refers to all possible output values (y-values). For quadratic functions, the domain is typically all real numbers, but the range is determined by the vertex and the direction of opening. Understanding these concepts is essential for determining whether the relation is a function and for identifying its characteristics.
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Domain & Range of Transformed Functions