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Graph the following and state the a, h, and k values that shift a graph: y = 2(x+3)²- 4 and please show work.

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Final Answer:

The quadratic function y = 2(x + 3)² - 4 represents a vertically stretched parabola with a factor of 2, shifted 3 units left, and 4 units downward from the standard parabola. The parameters are a = 2, h = -3, and k = -4, indicating the stretch/compression, horizontal shift, and vertical shift, respectively.

Explanation:

The given quadratic function, y = 2(x + 3)² - 4, is in vertex form, which is a concise representation of a parabolic equation. The values of a, h, and k in this form directly affect the graph's characteristics. The coefficient 'a' is the vertical stretch or compression factor, influencing the steepness of the parabola. In this case, a = 2, indicating a vertical stretch by a factor of 2 compared to the standard parabola y = x². As a result, the graph is narrower and opens upwards, intensifying the curvature.

The values of h and k represent horizontal and vertical translations, respectively. Here, h = -3 implies a horizontal shift of 3 units to the left, moving the vertex from the origin to (-3, 0). Additionally, k = -4 signifies a vertical shift of 4 units downward, relocating the vertex to (-3, -4). These transformations alter the position of the parabola without changing its shape.

In summary, the graph of the function is a vertically stretched parabola that has been shifted three units to the left and four units downward from the standard parabola. Understanding these parameters provides a comprehensive insight into how the equation shapes the graph.

Graph the following and state the a, h, and k values that shift a graph: y = 2(x+3)²- 4 and-example-1
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