Graphs and Transformations
Ever wondered why maths teachers obsess over graphs? They're basically the visual language of functions, and mastering them makes complex problems much easier to solve.
A cubic function follows the pattern f = ax³ + bx² + cx + d, where a ≠ 0. These create those distinctive S-shaped curves you've probably sketched before. Quartic functions take it up a notch with f = ax⁴ + bx³ + cx² + dx + e, where a ≠ 0, creating more complex curves with potentially multiple turning points.
Reciprocal graphs like y = k/x and y = k/x² have a special feature called asymptotes - invisible lines the curve approaches but never actually touches. For reciprocals, these asymptotes sit at x = 0 and y = 0, creating that distinctive hyperbola shape.
When it comes to transformations, think of them as ways to slide, flip, or stretch your original graph. Reflections are straightforward: y = -f flips across the x-axis, whilst y = f flips across the y-axis. Translations shift the entire graph: y = f + a moves it vertically by 'a' units, and y = f moves it horizontally by 'a' units (watch the direction - it's opposite to what you might expect!).
Stretches change the graph's size rather than position. The transformation y = af stretches vertically by factor 'a', whilst y = f(ax) stretches horizontally by factor 1/a. Remember, when you transform any function, the asymptotes move along with it - they're not fixed in place!
Key tip: Always sketch the original graph first, then apply transformations step by step. This prevents confusion when dealing with multiple transformations.