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.

---

Definitions:

Confirmation Bias

The tendency to search for, interpret, favor, and recall information in a way that confirms one's preexisting beliefs or hypotheses.

Wason Selection Task

A logic puzzle that demonstrates the difficulty people have in testing rules in a rational or logical way, often used in the study of deductive reasoning.

Propositional Rule

A statement in logic that expresses a relationship between concepts through propositions or declarative sentences.

Prefrontal Cortex

The part of the frontal lobes of the brain located at the front of the frontal lobe, involved in complex behaviors such as planning, decision making, and moderating social behavior.