Complete the Square to Determine Whether the Equation $y ^ { 2 } + 2 y - 12 x + 38 = 0$

question 47

Essay

Complete the square to determine whether the equation $y ^ { 2 } + 2 y - 12 x + 38 = 0$ represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

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Definitions:

Ecological Psychology

A branch of psychology focused on studying the relationship between organisms and their environment, emphasizing the importance of natural contexts for understanding behavior.

Behavior Settings

Specific places or contexts where a distinct and predictable pattern of behavior occurs, influenced by the physical and social environment.

Environmental Impact

The effect or influence of human activities on the natural environment, including pollution, habitat destruction, and climate change.

Territoriality

The behavior by which individuals or groups lay claim to and defend specific areas as their own.