Functions & Curve Sketching

10 methods · 25 questions

★ 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 (3)

STEP 3 2006 · Q1

Sketch the curve with cartesian equation

y = \frac{2 x ( x ^{2} - 5 )}{x ^{2} - 4}

and give the equations of the asymptotes and of the tangent to the curve at the origin.

Hence determine the number of real roots of the following equations:

$3 x (x^{2} - 5) = (x^{2} - 4) (x + 3)$ ;
$4 x (x^{2} - 5) = (x^{2} - 4) (5 x - 2)$ ;
$4 x^{2} (x^{2} - 5)^{2} = (x^{2} - 4)^{2} (x^{2} + 1)$ .

STEP 2 2011 · Q1

Sketch the curve $y = 1 - x + 3 + x$ . Use your sketch to show that only one real value of $x$ satisfies $1 - x + 3 + x = x + 1,$ and give this value.
Determine graphically the number of real values of $x$ that satisfy $2 1 - x = 3 + x + 3 - x .$ Solve this equation.

STEP 2 2013 · Q1

Find the value of $m$ for which the line $y = m x$ touches the curve $y = ln x$ . If instead the line intersects the curve when $x = a$ and $x = b$ , where $a < b$ , show that $a^{b} = b^{a}$ . Show by means of a sketch that $a < e < b$ .
The line $y = m x + c$ , where $c > 0$ , intersects the curve $y = ln x$ when $x = p$ and $x = q$ , where $p < q$ . Show by means of a sketch, or otherwise, that $p^{q} > q^{p}$ .
Show by means of a sketch that the straight line through the points $(p, ln p)$ and $(q, ln q)$ , where $e ⩽ p < q$ , intersects the $y$ -axis at a positive value of $y$ . Which is greater, $π^{e}$ or $e^{π}$ ?
Show, using a sketch or otherwise, that if $0 < p < q$ and $\frac{ln q - ln p}{q - p} = e^{- 1}$ , then $q^{p} > p^{q}$ .