To assess how the graph of the function has moved down three units from the parent function \( y = x^2 \), you must consider the transformation and key features of the new function, which can be expressed as \( y = x^2 - 3 \).
Here are some key features to help illustrate this transformation:
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Symmetry: The parabola retains its symmetry about the y-axis, just like the parent function \( y = x^2 \). This means that for every point \((x, y)\) on the graph, there exists a corresponding point \((-x, y)\).
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Direction of the Graph: The graph still opens upwards, as the coefficient of \( x^2 \) is positive in both the parent function and the transformed function \( y = x^2 - 3 \).
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Rate of Change: The rate of change, or the slope of the tangent lines at any given point on the graph, remains the same as it is determined by the derivative of the function. The shape of the graph has not changed, just its vertical position.
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Y-Intercept: To find the y-intercept of the new function, substitute \( x = 0 \): \[ y = 0^2 - 3 = -3 \] So, the y-intercept of the graph \( y = x^2 - 3 \) is at the point (0, -3), which is three units lower than the y-intercept of the parent function, which is (0, 0).
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Vertex: The vertex of the parent function is at the point (0, 0). Under the transformation \( y = x^2 - 3 \), the vertex moves down to (0, -3).
In summary, the key features that illustrate the downward movement of the graph by three units from the parent function include the y-intercept, vertex position, and the fact that the symmetry and direction (upward-opening parabola) remain unchanged.