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
The graph of $y = x^3 - 5x$ is drawn on the grid for values of $x$ from $-3$ to $3$ By drawing a suitable straight line on the grid, find estimates for the solutions of

$x^3 - 5x - 2 = 0$

(2 marks)
Level 4 · 2 marksTry it
2
The battery of a phone lasts $y$ hours when the brightness of its screen is set to $x$ %.

Here is a sketch of the graph for $20 \leqslant x \leqslant 100$ (a) Assume that the equation of the graph is $y = \dfrac{k}{x}$ where $k$ is a constant.

Work out the value of $y$ when $x = 50$

(2 marks)
(b) In fact, the battery lasts $9$ hours when the brightness is $100$ %.

What does this mean about the assumption in part (a)?

Tick one box.

☐ The assumption is correct

☐ The assumption is not correct

☐ It is not possible to tell

Show working to support your answer.

(2 marks)
Level 4 · 4 marksTry it
3
Here is a sketch of the curve $y = (x + 3)(x - 2)(x - 5)$ (a) Write down the coordinates of the point where the curve crosses the $y$ -axis.

(1 mark)
(b) Write down the values of $x$ where the curve crosses the $x$ -axis.

(1 mark)

Sam says,

" $y$ is negative for exactly two positive whole numbers $x$ ."
(c) Without expanding the brackets, show that Sam is correct.

(2 marks)
Level 4 · 4 marksTry it
4
The curve $y = \dfrac{6}{x}$ and the straight line $y = x + 1$ are drawn on the grid.(a) Use the graphs to solve $\dfrac{6}{x} = x + 1$

(2 marks)
(b) Show that the solutions of $\dfrac{6}{x} = x + 1$ are also the solutions of $x^2 + x - 6 = 0$

(2 marks)
Level 4 · 4 marksTry it
5
The graph of $y = x^3 - 12x$ is drawn on the grid for values of $x$ from $-4$ to $4$ (a) Write down the coordinates of the turning point of the graph that is a maximum.

(1 mark)
(b) Use the graph to find the number of solutions of $x^3 - 12x = 10$

(1 mark)
(c) The equation $x^3 - 12x = k$ has exactly one solution.

Write down a possible value of $k$ .

(1 mark)
(d) Write down the two values of $k$ for which $x^3 - 12x = k$ has exactly two solutions.

(1 mark)
Level 4 · 4 marksTry it