Cubic and reciprocal graphs

Foundation and Higher

The idea

A cubic such as $y = x^3$ has an S-shape. A reciprocal graph $y = \frac{1}{x}$ has two separate branches and never touches either axis, because $x = 0$ has no value.

Recognise the shape from the highest power: $x^2$ is a U, $x^3$ an S, $\frac{1}{x}$ two branches.
Plot from a table of values.
For a real-life graph, say what the gradient and the axes mean.

Watch out Never join the two branches of $y = \frac{1}{x}$ across $x = 0$ .

See it worked

Worked exampleNon-calculator2 marks

Here are four equations: $y = 2x$ , $y = x^2$ , $y = x^3$ and $y = \dfrac{1}{x}$ .

(a) Which equation has a graph that never touches either axis?

(b) Which equation has a graph passing through $(2, 8)$ and $(-2, -8)$ ?

(a) $\frac{1}{x}$ is never 0, and $x = 0$ is not allowed, so the reciprocal graph never meets either axis.
(b) $2^3 = 8$ and $(-2)^3 = -8$ .

Answer (a) $y = \frac{1}{x}$ ; (b) $y = x^3$

Your turn

Question 1Non-calculator3 marks

The time, $t$ hours, to empty a pool with $n$ identical pumps is given by $t = \dfrac{60}{n}$ .

(a) Work out the time taken with 4 pumps.

(b) How many pumps are needed to empty the pool in 5 hours?

Question 2Non-calculator4 marks

The curve has equation $y=x^3-3$ .

(a) Find the y-coordinate when x = −2.

(b) Find the change in y as x increases from 1 to 3.

Question 3Non-calculator4 marks

The graph of $y = \frac{12}{x}$ is drawn for $x > 0$ .

(a) Find y when x = 0.5.

(b) The line y = 3x crosses the curve. Find the coordinates of the crossing point.

In past papers

64 official questions in 59 papers since 2017. The latest:

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