Functions & Curve Sketching

11 methods · 29 worked questions using these methods

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

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To count or locate the real solutions of an equation depending on a parameter, rearrange it into the form $F (x) = ($ simple line or constant $)$ , sketch $y = F (x)$ once, then read off how many times a horizontal line $y = c$ or a sliding line $y = k x + \dots$ meets the curve as the parameter varies.

Trigger: "how many real roots", "for which values of $k$ ", or a transcendental equation that cannot be solved algebraically.

Instances:
(i) isolate a constant: $F (x) = c$ , and count crossings of the horizontal level $y = c$ against the turning points of $F$ .
(ii) split into two graphs $y = g (x)$ and $y = k x$ (or $y = m x + c$ ) and count their intersections as the line rotates or shifts.
(iii) the number of crossings changes exactly at a tangency of the line to the curve, which fixes the critical parameter value.

Linked questions for this method (3)

STEP 3 2006 · Q1

STEP 2 2011 · Q1

STEP 2 2013 · Q1