Students who have studied quadratic equations and the associated parabola curves will recognize the TotalCost vs distance plot. Using tools in the Algebra view they can construct a function of the form:

f(x) = w(x-h)²+k

than generates a graph that follows the trace of point P. There are multiple ways to accomplish this in GeoGebra. Below is one set of steps.

1. Use the View menu to deselect the Spreadsheet View and select the Algebra View.

2. Right-click on the Graphics 2 window and Zoom out to reduce the space required for the Wolfe Island image. Adjust the 3 view windows to show all entries in the Algebra View and maximize the Graphics window.

3. In the Graphics window adjust the x and y axes scales to have the TotalCost curve fill the graph.

4. Input: w = 1, k = 1, and h = 1. We will use these parameters in the general quadratic function and in GeoGebra all variable quantities must have initial values before being used.

5. Input: f(x) = w(x-k)^2+h  This will generate the function f(x) = 1(x-1)²+1

6. Adjust the values of the parameters w, k, and h to generate a quadratic function with a graph that passes through the trace of point P. To change the parameter values:

• In the Algebra View double left-click on a parameter
• In the window that appears enter the new parameter value
• Press the Enter key 

7. If students have been studying quadratic functions and their curves they should recognize the coordinates of the vertex as the values of k and h. From this they can identify the distance along the shoreline for the minimum cost and the least TotalCost.

f(x) = w(x-h)^2+k

applet with GeoGebra tools