Line Graphs

At a glance 10 rules

1. 1Change between points$\text{slope} = \dfrac{y_2 - y_1}{x_2 - x_1}$$(x, y) = (\text{time, value})$
2. 2Steepest is not highestThe steepest segment is the biggest jump between neighbouring points, not the highest point; if the question says 'change', falls count too.
3. 3Biggest rise, fastest growthA big rise on a high base can be a smaller percentage than a small rise on a low base; rank by new over old when the question says 'percentage'.
4. 4Average change per gap$\dfrac{\text{last} - \text{first}}{\text{number of gaps}}$$\text{6 readings make 5 gaps}$
5. 5Unequal gapsWhen the points are not evenly spaced (2001, 2011, 2016), divide each rise by the years it covers before comparing.
6. 6Gap between two linesThe gap is the difference at each point; the lines cross where the gap changes sign, and between points the gap closes at the difference of the slopes.
7. 7Index lines$\text{value} = \text{base value} \times \dfrac{\text{index}}{100}$$\text{lines can cross, values may not}$
8. 8Average yearly growth (CAGR)$\text{CAGR} = \left(\dfrac{\text{end}}{\text{start}}\right)^{1/n} - 1$$n = \text{years};\ 1.1^5 \approx 1.61,\ 1.2^3 \approx 1.73$
9. 9Same rate ahead$\text{next} = \text{last} \times \dfrac{\text{last}}{\text{previous}}$$\text{same rise: } 2 \times \text{last} - \text{previous}$
10. 10Running totalsA cumulative line's period value is $C_t - C_{t-1}$ ($C_t$ = total to the end of $t$); a running total turns positive only once the gains cover all earlier losses.