Sketching Cubics and Reciprocal Curves

Sketch cubic curves and the curves y = k/x and y = k/x² with their axis crossings and asymptotes, and use where two graphs meet to solve equations.

In the exam for Edexcel International AS Maths · 18 real past-paper questions below

Which way does a cubic go?

A sketch is not a careful drawing. It is a quick picture that shows the right shape and the key points. Marks go for the shape and for the key points: where it meets the axes, and any asymptotes.

Start with the shape. For a cubic, one number decides it: the number in front of $x^{3}$ .

Formula

y = a (x - p) (x - q) (x - r)

$a$ — the number in front of $x^{3}$ when you multiply out. Its sign sets the shape.
$p, q, r$ — the roots. The curve meets the x-axis at $x = p$ , $x = q$ and $x = r$ .

$a > 0$

Starts bottom left, ends top right. With three roots: up, down, up.

$a < 0$

Starts top left, ends bottom right. With three roots: down, up, down.

Why? For a big positive x, $x^{3}$ is huge and positive, so the curve is high on the right. For a big negative x, $x^{3}$ is huge and negative. A minus sign in front flips both ends.

Memory trick

Positive cubic: low to high, like a plane taking off. Negative cubic: high to low, like a plane landing.

Key idea

Look at the sign in front of $x^{3}$ first. It tells you which corner the curve starts in.

Sketching a factorised cubic

Most exam cubics come factorised, or nearly. Then the sketch takes four steps.

Steps

1
1. Factorise fully, if it is not done for you.
2. Roots: set each bracket to 0. Mark them on the x-axis.
3. y-intercept: put $x = 0$ . Mark it on the y-axis.
4. Shape: check the sign of $x^{3}$ , then join the points smoothly.

Worked example

1
Sketch $y = (x + 1) (x - 2) (x - 4)$ .
2
$x = - 1, 2, 4$ — Each bracket equal to 0. Crosses at (−1, 0), (2, 0), (4, 0).
3
$y = (1) (- 2) (- 4) = 8$ — Put $x = 0$ . Meets the y-axis at (0, 8).
4
$x \times x \times x = x^{3}$ — Positive, so it starts bottom left.
5
$up through - 1, down through 2, up through 4$ — Join the points with one smooth curve.

Worked example

1
Sketch $y = 9 x - x^{3}$ .
2
$y = x (9 - x^{2}) = x (3 - x) (3 + x)$ — Take out x, then use the difference of two squares.
3
$x = 0, 3, - 3$ — Crosses at (−3, 0), the origin and (3, 0).
4
$- x^{3}$ — Negative, so it starts top left and ends bottom right.
5
$down through - 3, up through 0, down through 3$ — The y-intercept is the origin, so label it O.

Mark schemes often give B1 for the shape, B1 for the x-axis points and B1 for the y-intercept. Write the coordinates on the sketch, like (0, 8), not just 8 floating nearby.

Example

A sketch also answers sign questions. For $y = (x + 1) (x - 2) (x - 4)$ , the curve is on or above the axis for $- 1 \leq x \leq 2$ and for $x \geq 4$ .

Common mistake

Sketching from the expanded form, like $- x^{3} + 9 x$ , without factorising. You cannot see the roots until you factorise.

Key idea

Factorise, find where it crosses, find the y-intercept, check the sign of $x^{3}$ . Then draw.

Repeated roots touch the axis

What if a bracket is squared, like $(x - 3)^{2}$ ? Then $x = 3$ is a repeated root. The curve comes down to the axis there, touches it, and turns back.

Rule

A single bracket: the curve crosses the x-axis. A squared bracket: the curve touches the x-axis and turns there.

Why? Near $x = 3$ , the squared bracket is never negative. So the curve does not change sign there. It stays on one side of the axis.

Worked example

1
Sketch $y = (2 x + 1) (x - 3)^{2}$ .
2
$2 x + 1 = 0 \Rightarrow x = - \frac{1}{2}$ — Single bracket. Crosses at $(- \frac{1}{2}, 0)$ .
3
$(x - 3)^{2} = 0 \Rightarrow x = 3$ — Squared bracket. Touches at (3, 0).
4
$y = (1) (- 3)^{2} = 9$ — Meets the y-axis at (0, 9).
5
$2 x \times x \times x = 2 x^{3}$ — Positive. Up through $- \frac{1}{2}$ , up through (0, 9), over a peak, down to touch at 3, then up again.

Example

Letters work the same way. For $y = (x - k) (x + 2)^{2}$ with $k > 0$ : crosses at $(k, 0)$ , touches at (−2, 0), meets the y-axis at $(0, - 4 k)$ .

Some questions turn this round: they show the graph and ask for the equation.

Worked example

1
A cubic touches the x-axis at $x = 3$ , crosses it at $x = - 1$ , and passes through (0, 18). Find its equation.
2
$y = a (x + 1) (x - 3)^{2}$ — Touch at 3 gives a squared bracket. Cross at −1 gives a single one.
3
$18 = a (1) (- 3)^{2} = 9 a$ — Put in the point (0, 18).
4
$a = 2, y = 2 (x + 1) (x - 3)^{2}$ — Do not forget a. Without it, the curve would pass through (0, 9).

Common mistake

Drawing the curve straight through a repeated root. Examiners check that it turns on the axis there and does not cross.

Key idea

Squared bracket: touch and turn. Single bracket: cross.

Reciprocal curves and asymptotes

You have met $y = \frac{k}{x}$ . Now compare it with $y = \frac{k}{x ^{2}}$ . Both have the axes as asymptotes, and both never touch them. Their shapes differ.

$y = \frac{k}{x}$ , $k > 0$

Two branches in opposite corners: top right and bottom left.

$y = \frac{k}{x ^{2}}$ , $k > 0$

Both branches above the x-axis: top right and top left. Each is the mirror image of the other in the y-axis.

Why? $x^{2}$ is positive for every $x \neq = 0$ . So $\frac{k}{x ^{2}}$ always has the sign of k. If k is negative, both branches drop below the x-axis.

Memory trick

The square keeps both arms on the same side. $\frac{k}{x ^{2}}$ with $k > 0$ is a ‘hands up’ graph: both arms up.

Exams often move the curve. Adding a number moves the horizontal asymptote. Changing x to $x + a$ moves the vertical one.

Worked example

1
Sketch $y = \frac{4}{x ^{2}} - 1$ .
2
$x = 0$ — You cannot divide by 0. Vertical asymptote.
3
$y = - 1$ — Far out, $\frac{4}{x ^{2}}$ is almost 0. Horizontal asymptote.
4
$\frac{4}{x ^{2}} = 1 \Rightarrow x^{2} = 4 \Rightarrow x = \pm 2$ — Crosses the x-axis at (−2, 0) and (2, 0).
5
$shape of \frac{4}{x ^{2}}, moved down 1$ — Both branches above $y = - 1$ . No y-intercept, since $x = 0$ is an asymptote.

Worked example

1
Sketch $y = \frac{2}{x + 3}$ .
2
$x + 3 = 0 \Rightarrow x = - 3$ — The bottom is 0. Vertical asymptote.
3
$y = 0$ — Far out, the fraction is almost 0. It never equals 0, as the top is 2. Horizontal asymptote.
4
$y = \frac{2}{0 + 3} = \frac{2}{3}$ — Meets the y-axis at $(0, \frac{2}{3})$ .
5
$shape of \frac{2}{x}, moved left 3$ — Branches top right and bottom left of the point where the asymptotes meet.

Marks usually go for the shape, the crossings and the asymptotes. Write each asymptote as an equation, such as $y = - 1$ . A dashed line with no label may not score.

Common mistake

Drawing $y = \frac{k}{x ^{2}}$ with a branch in the bottom left. For $k > 0$ , nothing is below the x-axis.

Key idea

The word comes from the Greek for ‘not falling together’. The Greek geometer Apollonius used it about 2200 years ago, in his work on curves, for lines that never meet the curve.

Key idea

Bottom = 0 gives the vertical asymptote. The number added on gives the horizontal one. Then find the crossings.

How many solutions? Count meetings

Some equations are too hard to solve by hand. But you can still say how many solutions they have.

Rule

The solutions of $f (x) = g (x)$ are the x-values where the graphs of $y = f (x)$ and $y = g (x)$ meet. Count the meeting points to count the solutions.

Worked example

1
Use sketches of $y = x (x - 3)^{2}$ and $y = \frac{1}{x ^{2}}$ to find how many real solutions $x (x - 3)^{2} = \frac{1}{x ^{2}}$ has.
2
$y = x (x - 3)^{2}$ — Positive cubic. Crosses at the origin, touches at (3, 0).
3
$y = \frac{1}{x ^{2}}$ — Both branches above the x-axis.
4
$x < 0$ — The cubic is below the axis and the reciprocal is above. No meeting.
5
$0 < x < 3$ — The cubic rises above the reciprocal, then falls back to 0 at $x = 3$ . Two meetings.
6
$x > 3$ — The cubic climbs for ever, the reciprocal sinks towards 0. One more meeting.
7
$3 solutions$ — Give the reason: the graphs meet 3 times.

To score, both sketches must be right. The reason must be about where the graphs meet each other, not where they cross the axes.

A ‘show that’ part often asks for the equation those meeting points satisfy. Set the two equal and clear the fraction, one line at a time.

Worked example

1
Show that the x-coordinates where the curves meet satisfy $x^{5} - 6 x^{4} + 9 x^{3} - 1 = 0$ .
2
$x (x - 3)^{2} = \frac{1}{x ^{2}}$ — Set the two curves equal.
3
$x^{3} (x - 3)^{2} = 1$ — Multiply both sides by $x^{2}$ .
4
$x^{3} (x^{2} - 6 x + 9) = 1$ — Expand the square.
5
$x^{5} - 6 x^{4} + 9 x^{3} - 1 = 0$ — Multiply out and collect on one side. This matches the given answer.

Common mistake

Jumping straight to the printed answer. In a ‘show that’, write every step, or you can lose all the marks.

Key idea

Number of solutions = number of times the graphs meet. Always say why.

Which method? Read the question

Each part tells you what to show. Match the wording to the tool.

You see	You do
A cubic, not factorised	Factorise fully first
“Showing the coordinates where it meets the axes”	Roots on the x-axis; put $x = 0$ for the y-intercept
A squared bracket	Touch the axis and turn there
$\frac{k}{x}$ or $\frac{k}{x ^{2}}$ with numbers added	Find both asymptotes and write them as equations
“State the number of roots, with a reason”	Count where the two graphs meet; say so
“Find the equation of the cubic” from a graph	Write $a (x - p) (x - q) (x - r)$ ; use a point to find a
“In terms of k”	Leave the letter in your coordinates

Common mistake

Leaving out the y-intercept. It is a mark of its own, and it also checks your shape.

Key idea

Shape from the sign, points from the factors, asymptotes as equations. Label everything on the sketch.

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