GCSE · Graphs of functions

Cubic and reciprocal graphs (tables, plotting and asymptotes)Exam-style questions

Draw y = x³ − 3x and y = 4/x from a table of values (a cube keeps its sign, and there is no point at x = 0), join the points with one smooth curve (two separate branches for a reciprocal), read the solutions of an equation off the curve, and know the shapes of ±x³ and ±k/x and the asymptotes they never touch.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
(a) Complete the table of values for $y = x^3 - 2x + 1$
$x$ $-2$ $-1$ $0$ $1$ $2$
$y$ $-3$ $1$
(2 marks)
(b) On the grid, draw the graph of $y = x^3 - 2x + 1$ for values of $x$ from $-2$ to $2$ (2 marks)
(c) Use your graph to find estimates for the solutions of the equation

$x^3 - 2x + 1 = 2$

(2 marks)
Level 3 · 6 marksTry it
2
(a) Complete the table of values for $y = x^3 - 2x^2 - 3x$
$x$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$
$y$ $0$ $0$ $-6$ $0$
(2 marks)
(b) On the grid, draw the graph of $y = x^3 - 2x^2 - 3x$ for values of $x$ from $-2$ to $4$ (2 marks)
(c) Use your graph to find estimates for the solutions of the equation

$x^3 - 2x^2 - 3x = -2$

(2 marks)
Level 3 · 6 marksTry it
3
Here is a table of values for $y = x + \dfrac{4}{x}$
$x$ $-4$ $-2$ $-1$ $-0.5$ $0.5$ $1$ $2$ $4$
$y$ $-5$ $-4$ $-5$ $-8.5$ $8.5$ $5$ $4$ $5$
(a) Draw the graph of $y = x + \dfrac{4}{x}$ for $-4 \le x \le 4$ , $x \ne 0$ (3 marks)
(b) Use your graph to solve $x + \dfrac{4}{x} = 4.5$

Give your answers correct to 1 decimal place.

(2 marks)
(c) Explain how the graph shows that the equation $x + \dfrac{4}{x} = 3$ has no solutions.

(1 mark)
Level 3 · 6 marksTry it
4
Ravi tries to sketch the graph of $y = x^3 - 8$

Here is his sketch.Make two different criticisms of his sketch.

Criticism 1 $\;$ ……………………………………

Criticism 2 $\;$ ……………………………………

(2 marks)
Level 3 · 2 marksTry it
5
Here are six graphs.

ABCDEFWrite down the letter of the graph that could have the equation
(a) $y = -x^3$

(1 mark)
(b) $y = \dfrac{2}{x}$

(1 mark)
(c) $y = x^3 - 4x$

(1 mark)
Level 3 · 3 marksTry it

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-3$		$1$

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$		$0$	$0$		$-6$	$0$

$x$	$-4$	$-2$	$-1$	$-0.5$	$0.5$	$1$	$2$	$4$
$y$	$-5$	$-4$	$-5$	$-8.5$	$8.5$	$5$	$4$	$5$

2 more questions on cubic and reciprocal graphs (tables, plotting and asymptotes)

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