Sketching curves

The idea

A sketch shows the shape and where the curve meets the axes. For a polynomial in factorised form, the factors give the roots: a squared factor touches the axis, a single factor crosses it. $y = \frac{k}{x}$ has asymptotes $x = 0$ and $y = 0$ .

Find the $y$ -intercept by putting $x = 0$ .
Find the roots from the factors and decide cross or touch.
Use the leading term for the ends: a positive cubic rises to the right.

Watch out Label where the curve meets the axes and any asymptotes with their equations.

See it worked

Worked example

Describe the sketch of $y = (x - 1)^2(x + 3)$ .

Roots: $x = 1$ (squared, so it touches) and $x = -3$ (crosses).
$y$ -intercept: $(0 - 1)^2(0 + 3) = 3$ .
It is a positive cubic: low on the left, high on the right.

Answer Crosses at $(-3, 0)$ , touches at $(1, 0)$ , meets the $y$ -axis at $(0, 3)$

Your turn

Question 1Non-calculator3 marks

Where does $y = (x + 1)(x - 2)(x - 5)$ meet the $y$ -axis?

Question 2Non-calculator3 marks

Write down the equations of the asymptotes of $y = \frac{3}{x} + 2$ .

Question 3Non-calculator3 marks

The curve $y = \frac{k}{x}$ passes through $(2, 5)$ . Find $k$ .

In past papers

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