You may have experience with connecting the dots from your childhood.
Connecting the dots is still an important skill, especially in math and science.
Often, it is necessary to find a function that is known to pass through pre-assigned points in the coordinate system. There are so many different combinations of functions that it is impossible to determine mathematically which of those combinations dictate how a set of data points are behaving.
That choice is often made based on the user’s knowledge of the source of the data. Once a particular type of function is selected, it is the role of mathematics to determine how they are combined to best represent the data points.
This simulation allows you to experiment with one important type of mathematical dot-connecting, using polynomials that pass through the dots, or points in the coordinate system.
Enter up to ten points by typing their x- and y-values into the blank ordered pairs.
Enter x-intercepts by using a 0 for the y-value of the point, and enter y-intercepts by using 0 for the x-value.
When you click the “GO” button, the simulation will calculate the coefficients of the polynomial of smallest degree that passes through all of the point you entered.
The computed polynomial will be displayed at the bottom of the screen,
… and the points and polynomial will be shown in the graph area.
There are two rules to keep in mind: One, all of the x-values must be different,
and Two, at least one y-value must be different from zero.
You can change any point after the computation by re-entering the coordinates. Just remember to click the “GO” button when you have finished making changes.
After the polynomial is computed, you can find the y-value associated with any x-value by using the “x =” box beneath the points. Type the value by hand or use the slider to set the value of x, then click the “Calculate” button to get the corresponding y-value. The point will be displayed as an open circle on the graph.
Click the “Clear” button to start all over.
You will probably find that the graphs stretch off the top and bottom of the graph area, Use the zoom control to zoom vertically, so that more or less of the y-axis appears in the window. The x-axis will remain fixed. You can also use the scroll wheel on your mouse.
You can add points to the list by clicking in the graph area, but more accuracy can be had by entering the coordinates manually.
Try one of the exercises by entering the data both ways, once by clicking and once by typing. Small errors can lead to large changes in the results, so be careful! And try the rest of the exercises too! Good luck!