The Use of Graphing Software in the Mathematics Classroom (2)

Autograph software has 2D and 3D graphing capabilities for pre-calculus topics such as graphing, transformations, inverse functions, trigonometric functions, conic sections, vectors, slopes; calculus topics such as functions and integrals; and topics in statistics. Autograph excels at visualizing mathematical concepts. A key strength of this software is that parameters can be included in mathematical equations and functions and varied dynamically to observe their effects on the graphs, helping students gain a deep understanding of mathematical properties [13]. For example, the parameters \( a \), \( h \), and \( k \) in the quadratic function \( f \) given by:

\[ f(x) = a(x - h)^2 + k \]

can be varied one at a time while exploring the resulting changes in the graph. Such an experiment can be used to investigate the coordinates of the vertex of the graph of function \( f \) and its range.

Task: Students explore the relationship between the sine function as defined in the unit circle and its graph on rectangular axes. This allows students to view the sine function using two complementary representations.

Fig. 1: Sine function in unit circle and rectangular system of axes.

In trigonometry, an angle is defined as a measure of rotation. Autograph is dynamic graphing software that can easily be used to explain the concept of angles and the relationship between an angle in standard position and its associated trigonometric functions. This activity allows students to explore the period, range, and five key points over one period of the sine function.

Task: Students explore the graphs of both sides of equations and use definitions to decide whether a given equation is conditional or an identity. Examples used in classes include:

• \( 2x = 4 \)
• \( (x + 2)^2 = x^2 + 4x + 4 \)
• \( -2x^2 + 3 = x \)
• \( \sin\left(x + \dfrac{\pi}{3}\right) = \sin(x) + \sin\left(\dfrac{\pi}{3}\right) \)
• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( \cot(x) (\sec(x) + \tan(x)) = \dfrac{1}{\sin(x)} + 1 \)

Fig. 2: Verifying identities.

This activity allows students to explore graphically the difference between a conditional equation and an identity. Questions on the definition of an identity posed to students before and after this activity showed that students gained a deeper understanding of the concept of identities.