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
It is given that

$\mathrm{f}(x) = x(x + 4)(b - x)$

where $b$ is a positive constant.

Sketch the graph of $y = \mathrm{f}(x)$

(3 marks)
Level 2 · 3 marksTry it
2
The line $y = 5 - x$ meets the curve $y = \dfrac{6}{x}$ . Find the coordinates of the points of intersection.
Level 3 · 3 marksTry it
3
Figure 1

The curve $C$ , shown in Figure 1, has equation $y = \text{g}(x)$ , where $\text{g}$ is a cubic function.

The curve passes through the origin $O$ , and its turning points are at $(1,\ -7)$ and $(5,\ 25)$ .
(a) Write down the set of values of $x$ for which $\text{g}'(x) < 0$

(1 mark)

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

(2 marks)
Level 3 · 3 marksTry it
4
The curve $C_1$ has equation

$y = 3x^2 + 2 \qquad x > 0$

The curve $C_2$ has equation

$y = \frac{36}{x} - 4 \qquad x > 0$
(a) Verify that the curves intersect at $x = 2$

(2 marks)
(b) Write down the $y$ coordinate of the point of intersection.

(1 mark)
Level 3 · 3 marksTry it
5
Figure 1

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

The curve passes through the origin $O$ and has stationary points at $(-2,\ 14)$ and $(4,\ -40)$ .
(a) Write down the set of values of $x$ for which $\text{f}'(x) < 0$

(1 mark)

The line $l$ has equation $y = k$ , where $k$ is a constant.

Given that $l$ meets $C$ at exactly one point,
(b) find the set of values of $k$ , giving your answer in set notation.

(2 marks)
Level 3 · 3 marksTry it

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