Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex Form of a Parabola
The vertex form of a parabola is expressed as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for identifying the vertex directly and understanding the transformations applied to the basic parabola y = x². The parameter 'a' determines the width and direction of the parabola, with positive values opening upwards and negative values opening downwards.
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Effect of 'a' on Parabola Shape
The coefficient 'a' in the vertex form of a parabola affects its width and direction. If |a| > 1, the parabola is narrower than the standard parabola, while |a| < 1 makes it wider. Additionally, if 'a' is positive, the parabola opens upwards, indicating a minimum point, whereas a negative 'a' indicates it opens downwards, signifying a maximum point. This concept is crucial for maintaining the same shape as the given parabolas.
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Finding Vertex from Given Conditions
To write the equation of a parabola in vertex form based on given conditions, one must identify the vertex coordinates. In this case, the maximum is given as 4 at x = -2, which means the vertex is (-2, 4). This information allows us to substitute h and k into the vertex form equation, ensuring that the new parabola has the desired maximum while retaining the shape defined by the original parabolas.
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