Function graphs

Introduction

There are four ways to create graphs in the Geometry & Graphs question type:

1. Plotting a formula
2. Freehand drawing a curve through the Freehand Graphs tool.
3. Using a standard graph
4. Drawing a polynomial through a set of points with the polynomial tool.

Plotting a formula

Use the Formula button in the bottom line of the toolbar to enter the formula you want to display. The expected form is $y=expression$, where $expression$ depends on $x$.

Figure: A graph used as part of a construction to illustrate the concept of finite differences and tangents.

You can work with the graph like any other element in the canvas. For instance, you can adjust the style of the graph line in the element panel. You can also position glider points on the graph or put a point on the intersection of the graph with other elements.

Freehand Graphs Tool

When students first encounter function graphs, it is important to practice drawing graphs by hand. By doing so, students learn the connections between the symbolic formula and the graph's shape.

The Freehand Graphs tool allows you to draw a smooth curve through a sequence of points. We discussed this tool in the Geometrical Constructions section.

Figure: A graph created with the Freehand Graphs tool. You can drag the points or use the Freehand Graphs tool to add additional points to the graph.

Standard Graphs

Algebrakit also supports graphing standard relations using the Standard Graphs tool. This tool provides a quick, easy way to draw and evaluate such graphs. Standard graphs are ideal for exercises involving function transformations or reasoning about the properties of functions, such as minimums, maximums, asymptotes, amplitudes, and equilibria. You can draw each standard graph by defining two points that lie on the graph, or an asymptote and one or two points where applicable.

You can use the Standard Graphs tool to draw the following functions:

• Parabolas
• Hyperbolas
• Sines and Cosines
• Tangents
• Roots
• Logarithms
• Exponential Functions

Figure: Several standard graphs. You can drag the points to edit the graphs, or use the various standard graph tools to add new graphs.

Another advantage of standard graphs is that you can use them to evaluate exercises easily. A guide to evaluating standard graphs can be found here.

Polynomial tool

With the Polynomial tool, you can draw polynomials of any degree. The Polynomial tool interpolates $n$ points into a polynomial with a degree of $n-1$.

Figure: A third-degree polynomial created with the Polynomial tool. You can drag the points to edit the polynomial, or use the Polynomial tool to add a new polynomial.

Figure: A third-degree polynomial with two hidden fixed points and two free points.