17. Tangents, Stationary Points and Curve Sketching

Starting point: Years 9 and 10

Reading a graph

Look at any graph from left to right.

It is increasing where it goes up.
It is decreasing where it goes down.
A turning point is where it changes direction. At the very top or bottom it is flat for an instant.

For a parabola $y = a x^{2} + b x + c$ , the turning point is at $x = - \frac{b}{2 a}$ . For $y = x^{2} - 6 x + 5$ the turning point is at $x = 3$ , $y = - 4$ , and it is a minimum.

Equation of a line through a point

You will often know a gradient and a point. Use $y - y_{1} = m (x - x_{1})$ .

y = x³ − 3x. The tangent is flat at the maximum and the minimum.

Using the derivative

Tangent and normal

Gradient m = f'(a)
Normal gradient -1/m

Increasing, decreasing

f'(x) > 0 rising
f'(x) < 0 falling

Stationary points

f'(x) = 0
Max, min, inflection

Second derivative

f''(x) > 0 concave up
f''(x) < 0 concave down
Sign change: inflection

Sketch

Intercepts
Stationary points
Ends

Year 11Increasing, decreasing and stationary points

For $y = x^{3} - 3 x$ : $\frac{d y}{d x} = 3 x^{2} - 3 = 3 (x - 1) (x + 1)$ .

Stationary points where $3 x^{2} - 3 = 0$ : $x = \pm 1$ . Points $(- 1, 2)$ and $(1, - 2)$ .
$\frac{d y}{d x} < 0$ between $- 1$ and $1$ , so the curve decreases there.
$\frac{d y}{d x} > 0$ for $x < - 1$ and $x > 1$ , so it increases there.

A table of gradients shows the type of each point:

$x$	$- 2$	$- 1$	0	1	2
$\frac{d y}{d x}$	9	0	$- 3$	0	9
slope	up	flat	down	flat	up

Up, flat, down means a maximum at $x = - 1$ . Down, flat, up means a minimum at $x = 1$ .

Year 12The second derivative and concavity

Differentiate twice to get $f^{''} (x)$ , the rate of change of the gradient.

$f^{''} (x) > 0$ : concave up, curving like a cup.
$f^{''} (x) < 0$ : concave down, curving like a cap.

Testing stationary points. At a point where $f^{'} (a) = 0$ :

$f^{''} (a) > 0$ means a minimum.
$f^{''} (a) < 0$ means a maximum.
$f^{''} (a) = 0$ : use a table of gradients.

A point of inflection is where concavity changes, so $f^{''} (x) = 0$ and changes sign. For $y = x^{3} - 3 x$ , $f^{''} (x) = 6 x$ , which changes sign at $x = 0$ . The point $(0, 0)$ is an inflection.

A horizontal point of inflection has $f^{'} (a) = 0$ too, like $y = x^{3}$ at the origin.

Sketching checklist: domain, intercepts, stationary points and their nature, inflections, and what happens as $x \to \pm \infty$ .

Global maximum. On a closed interval, compare the stationary values with the values at the endpoints. The largest is the global maximum.

Quick summary

Increasing where the gradient is positive. Decreasing where it is negative.
Stationary points are where $f^{'} (x) = 0$ .
Tangent gradient $f^{'} (a)$ . Normal gradient $- \frac{1}{f ^{'} ( a )}$ . Use $y - y_{1} = m (x - x_{1})$ .
Use a table of gradients or $f^{''} (a)$ to tell a maximum from a minimum.
Concave up: $f^{''} > 0$ . Concave down: $f^{''} < 0$ . Inflection: $f^{''}$ changes sign.

Quiz

14 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.