Thus, the positive/negative points of the microbiological quality of the productive strata were identified via the joint analysis of the two-dimensional planes PC1 x PC2 and PC1 x PC3, which resulted in technical assistance more specific than that from using the first Cartesian plane decision-making method.

In this sense, at an industrial level, the use of reward and penalty systems to identify the main problems of raw material quality is possible with assistance from the data gleaned from the groupings placement in the Cartesian plane. This method would facilitate the logistics of capturing milk with quality chemical and microbiological characteristics, representing a dilution in production costs, as these indicators are directly related to the industrial yield.

They came up with drawings such as a butterfly when they discussed plotting points on a

Cartesian plane.

Given a set of coordinate data on a

Cartesian plane, Excel and other spreadsheets offer a simple yet powerful means of mapping the data, checking for data errors, and performing geometric calculations to find perimeters and areas.

To illustrate many of the features discussed above for cubic polynomials with real coefficients plotted in the Cartesian plane, consider the curve y = [x.sup.3] + 3[x.sup.2] + x - 5 shown in Figure 2 over the range -4 [less than or equal to] x [less than or equal to] 2.

The complex conjugate roots do not correspond to the locations of either turning point or the PoI shown in Figure 1 and have often been misplaced by well-meaning but ingenuous authors like Stroud (1986, see Programme 2, Theory of Equations, where he [incorrectly] places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation).

Part (a) required students to think about the 'height of the platform' in terms of the

Cartesian plane on which the diagram was set.

Note also that although each of the points of intersection is found by considering points of intersection of equations (1) and (2) in the Cartesian plane, these points correspond directly to the roots in the complex plane.

The quadratic equation with complex coefficients can readily be solved by considering the intersection of two hyperbolas in the Cartesian plane. The solutions lie in the complex plane.

Meanwhile, in the

Cartesian plane, a closely related topic deals with the solution of polynomials (ACARA, n.d., Unit 2, Topic 3: Real and Complex Numbers).

This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not make the full visual connection between the

Cartesian plane and the Argand plane that Bardell's three dimensional surfaces illustrated so well.

Australian secondary school students first encounter Equation (1) in Year 9 (i.e., Australian Curriculum ACMNA296), and subsequently explore its various features using techniques such as two-dimensional plots in the

Cartesian plane, factorisation, completing the square, and the general root formula shown in Equation (3).

The game is played by two participants on a grid that mimics the x- and y-coordinates of a

Cartesian plane and can be played in any classroom.