An example of a wine glass design, using three equations, is given below. The base of the wine glass is defined by a quadratic equation, the stem by a linear equation and the body of the glass by a square root function. The difficult part of the task(outlined on the next page) is determining the correct coefficients in the equations so that the parts of the glass fit together. The equations must be accompanied by domain statements so that only the relevant section of the graph is included in the design.

Your task
  Design a wine glass that uses at least 5 different equations to form the outline of the shape. At least half of your equations must be curves. It is easier if the design is done with the glass on its side, that is, with the axis of the design running along
the x-axis.
  Produce an accurate graph of your design. A sketch of the final product would be
appropriate.
  Submit a list of the equations you have used in the design. Include any domain restrictions.
  Be sure to include all working and justification in the development of your design.

To design a wine glass using a variety of equations, you can consider different aspects of the glass's shape such as the base, stem, body, and rim. Here's an example of a design using at least 5 different equations:

Equation 1: Base of the wine glass (Quadratic Equation)
Consider a quadratic equation that represents the shape of the base of the wine glass. The equation could be:
y = ax^2 + bx + c
You can choose specific values for a, b, and c to create the desired shape of the base. For example, you can set a = 0.01, b = 0, and c = 0.

Equation 2: Stem of the wine glass (Linear Equation)
To represent the stem, you can use a linear equation:
y = mx + d
Choose appropriate values for m and d to create the desired shape and height of the stem. For instance, you can set m = 0.2 and d = 2.

Equation 3: Body of the wine glass (Square Root Function)
To represent the body of the glass, which typically curves inward, you can use a square root function:
y = sqrt(x)
However, only a certain portion of the graph represents the body. Therefore, you need to set a domain restriction to include only the relevant section. For example, you can set the domain as x ≥ 0.

Equation 4: Contour of the glass (Curve Equation)
For added curvature and complexity, you can use a curve equation that represents the contour of the glass's body. This equation can be any non-linear function that creates the desired shape. For example, you can use a sine function:
y = sin(x)
Again, you can include a domain restriction, such as -π/2 ≤ x ≤ π/2, to limit the relevant section.

Equation 5: Rim of the glass (Circle or Ellipse Equation)
To represent the rim of the wine glass, you can use either a circle or an ellipse equation, depending on the desired shape. For example, if you want a circular rim, the equation could be:
(x - h)^2 + (y - k)^2 = r^2
Choose specific values for h, k, and r to create the size and position of the rim.

Remember to adjust the coefficients and constants in all equations to match the desired shape and proportions of the wine glass. It's helpful to plot these equations on a graph with an appropriate scale and coordinate system to visualize the final design accurately.

Once you have determined all the equations with their respective domain restrictions, you can graph them using software like GeoGebra or any graphing calculator. Plotting the equations will provide you with the outline of the wine glass design.

Please note that the specific equations and their parameters provided above are just examples. You are encouraged to adjust them according to your own design preferences and guidelines given in your task description.