As a CAD drafting trainee, you are using coordinate geometry to map out a hyperbolic architectural feature. Your instructor provides the equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$. Arrange the standard graphing procedure into the correct sequence to recall how to properly construct this hyperbola on your drafting grid.

In an urban planning project, a designer is using a hyperbolic curve to model a decorative walkway. The curve is defined by the equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$ in a coordinate system where units are in feet. To properly align the surrounding landscape features, the designer needs to identify the center of this hyperbola. What are the coordinates of the center (h, k) for this walkway?

As an acoustics engineer, you are designing a concert hall where the wall curvature follows a hyperbola. The cross-section of the wall is modeled by the equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$. To correctly place the seating and sound panels, you must identify the key geometric properties of this curve. Match each property with its correct value or description.

A technician at a precision manufacturing firm is using a computer-aided design (CAD) program to model a hyperbolic part. The part's profile is defined by the equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$. True or False: Because the term containing $x$ is the positive term in this equation, the hyperbola opens horizontally (to the left and to the right) rather than vertically.

Locating Vertices for a Hyperbolic Structural Component

A manufacturing technician is setting up a CNC machine to cut a hyperbolic template for a reflector. The profile of the template is modeled by the equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$. When laying out the design, the technician marks the center at (1, 2) and sketches a central guide rectangle. Since $a = 3$, they draw the rectangle's horizontal boundaries by moving 3 units left and right from the center. To define the vertical boundaries of this guide rectangle, the technician must move ____ units above and below the center.

Graphing Guide for a Hyperbolic Curved Acoustic Track