Example: Graphing $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$

Example: Graphing $\frac{(y+2)^2}{9} - \frac{(x+1)^2}{4} = 1$

Example: Graphing $4x^2 - 9y^2 - 24x - 36y - 36 = 0$

As an entry-level CAD drafter at Orbital Aerospace, you are tasked with modeling a hyperbolic reflector for a new satellite communication dish. The reflector's cross-section is modeled by a hyperbola offset from the origin at a center point $(h, k)$. To manually verify your drafting software's output, you need to sketch the hyperbola's framework. Arrange the standard mathematical steps for graphing this hyperbola in the correct order.

In civil engineering, cooling towers of power plants often feature a hyperbolic profile to maximize structural stability. An engineer is reviewing a blueprint where the cross-section of a tower is defined by the standard equation $\frac{(y - 5)^2}{64} - \frac{(x + 3)^2}{36} = 1$. According to the standard graphing procedure, which coordinates represent the center $(h, k)$ of this hyperbola?

In acoustics engineering, hyperbolic surfaces are used to design specialized sound-focusing reflectors. To verify a design's accuracy, a technician must identify the key components used in the standard graphing procedure for a hyperbola with a center at $(h, k)$. Match each component with its specific role or definition.

Procedural Framework for Hyperbolic Design

A manufacturing technician is following the standard six-step procedure to graph a hyperbolic path centered at $(h, k)$. According to the procedure, to determine if the transverse axis is horizontal or vertical, the technician must identify which variable's squared term is ____ in the standard form equation.