Topic 2 Rectangular (Cartesian) coordinates
Figure 2.1: Cartesian coordinate system
\(x\) and \(y\) are called the Cartesian (or rectangular) coordinates of \(P\).
The whole plane is split by the coordinate axes into four regions called quadrants.
Figure 2.2: Line graph
The graph of a function of the form \(f(x)=mx+c\) is a line; \(m\) is called the slope; and \(c\) is the intercept of the line. Note: \(c=f(0)\).
If \(P_1=(x_1,y_1)\) and \(P_2=(x_2,y_2)\) are any two distinct points on the line, then \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
2.1 Guessing a law from an experiment
Experimental data for two (physical) variables \(x,y\) can be plotted as points in the \((x,y)\)-plane. Sometimes they may look like they roughly lie on a line. This would indicate a linear law between \(x\) and \(y\), i.e. \(y=mx+c\) for some numbers \(m\) and \(c\). The constants can be measured from the plot (explained in a video).
For an example, see Table 2.1 and Figure 2.3.
| t (s) | v (m/s) |
|---|---|
| 1 | 7.7 |
| 2 | 10.5 |
| 3 | 13.3 |
| 4 | 15.5 |
| 5 | 16.3 |
| 6 | 20.5 |
| 7 | 23.0 |
Figure 2.3: Plotted v against t
Note: sometimes the plotted data may exhibit a different “shape”. It is useful to learn to “recognise” at least quadratics, exponentials and sine/cosine.
Extra Note (non-examinable): Nowadays one would use statistics/linear algebra to find the “best line” that fits the measured data. The most commonly used method for this is called linear regression (this is what the blue line in Figure 2.3 actually represents).