A-level · Algebra & functions

Sketching curvesExam-style questions

Drawing a curve from its equation without plotting points. Where it meets the axes, read from the factors; which way the ends go, from the leading term; whether it crosses or touches at each root; reciprocal curves and their asymptotes; finding where two curves meet; and counting meeting points from a sketch or with the discriminant.

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17 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The function f is defined by

$\mathrm{f}(x) = x(x + a)(x + 5) \quad \text{for } x \in \mathbb{R}$

where $a$ is a constant and $0 < a < 5$
(a) Sketch the graph of $y = \mathrm{f}(x)$

(3 marks)
(b) Sketch the graph of $y = \mathrm{f}(-3x)$

(2 marks)
Level 4 · 5 marksTry it
2
(a) Sketch the graph of

$y = \frac{1}{x^3}$

(2 marks)
(b) The graph of $y = \dfrac{1}{x^3}$ is to be transformed into the graph of $y = \dfrac{64}{x^3}$ using one stretch.

Rhys says that this can be done with a stretch parallel to the $y$ -axis.

Isla says that this can be done with a stretch parallel to the $x$ -axis.

State whether Rhys is correct, Isla is correct, both are correct or neither is correct, giving reasons for your answer.

(3 marks)
Level 4 · 5 marksTry it
3
Figure 1

Figure 1 shows a sketch of the curve $C$ with equation $y = \text{f}(x)$ , where $\text{f}(x)$ is a cubic polynomial.

The curve passes through the origin $O$ . The point $A(4,\ 32)$ is a maximum point of $C$ and the point $B(12,\ 0)$ is a minimum point of $C$ .
(a) Write down the set of values of $x$ for which $\text{f}'(x) < 0$

(1 mark)

The line $l$ , with equation $y = k$ where $k$ is a constant, meets $C$ at one point only.
(b) Find the set of values of $k$ , giving your answer in set notation.

(2 marks)
(c) Find the equation of $C$ . You may leave your answer in factorised form.

(3 marks)
Level 4 · 6 marksTry it
4
Figure 1

Figure 1 shows a sketch of the curve $C$ with equation $y = \text{h}(x)$ , where $\text{h}(x)$ is a polynomial of degree 3.

The curve has a minimum point at $P(-6,\ -18)$ and a maximum point at the origin $O$ .
(a) Write down the set of values of $x$ for which $\text{h}'(x) < 0$

(1 mark)

Given that the curve $C$ and the line with equation $y = k$ , where $k$ is a constant, have exactly one point in common,
(b) find the set of values of $k$ , giving your answer in set notation.

(2 marks)
(c) Find the equation of $C$ . You may leave your answer in factorised form.

(3 marks)
Level 4 · 6 marksTry it
5
The curve $C$ has equation $y = \dfrac{x + 2}{x - 1}$ .
(a) Write down the equations of the vertical and horizontal asymptotes of $C$ . (2 marks)
(b) Find the coordinates of the points where $C$ crosses the coordinate axes. (2 marks)
(c) The line $y = x$ intersects $C$ . Find the exact $x$ -coordinates of the points of intersection. (4 marks)
(d) State the number of points of intersection of $C$ with the line $y = x$ . (1 mark)
Level 4 · 9 marksTry it

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