A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 4: Graphs and transformations · Lesson 1

Cubic and quartic graphs

71 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-cubic-and-quartic-graphs/

Date

Do Now

Quick questions to start.

A1
Needed today
Factorise completely $5 x^{3} - 45 x$ .
[2 marks]
Answer
A2
Needed today
Expand and simplify $(x - 2) (x + 3) (x + 1)$ .
[2 marks]
Answer
A3
Last lesson
Find the set of values of $x$ for which $x^{2} < 3 x + 10$ .
[2 marks]
Answer
A4
From chapter 2
Find the coordinates of the turning point of $y = x^{2} - 8 x + 19$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Sketch $y = (x + 4) (x - 1) (3 - x)$ , showing where it meets the axes.

Hint Put

y = 0

, then

x = 0

. The

x^{3}

term is

x \times x \times (- x) = - x^{3}

B1
Sketch $y = (x - 2) (x + 1) (x + 5)$ , showing the coordinates of the points where it meets the axes.
[3 marks]

Example 2

Factorise $3 x^{3} + 12 x^{2} + 12 x$ completely. Hence sketch $y = 3 x^{3} + 12 x^{2} + 12 x$ .

Hint Take out the common factor

3 x

first. A squared factor gives a repeated root: the curve touches the axis there.

B2
Factorise $20 x^{2} - 4 x^{3} - 25 x$ completely. Hence sketch $y = 20 x^{2} - 4 x^{3} - 25 x$ , showing where it meets the axes.
[4 marks]

Example 3

Sketch $y = - (x + 2) (x - 1)^{2} (x - 3)$ , showing where it meets the axes.

Hint A quartic. The

x^{4}

coefficient is

- 1

, so both ends go down. The squared factor gives a point where the curve touches.

B3
Sketch $y = (x + 3) (x + 1) (x - 2)^{2}$ , showing the coordinates of the points where it meets the axes.
[3 marks]

Example 4

The sketch shows the curve $y = a x^{3} + b x^{2} + c x + d$ . It crosses the $x$ -axis at $x = - 1$ , $x = 2$ and $x = 5$ , and the $y$ -axis at $(0, - 20)$ . Find $a$ , $b$ , $c$ and $d$ .

Hint Write

y = a (x + 1) (x - 2) (x - 5)

and use the

y

-intercept to find

a

B4
The sketch shows the curve $y = x^{4} + b x^{3} + c x^{2} + d x + e$ . It crosses the $x$ -axis at $x = - 3$ and $x = 2$ , and touches it at $x = 1$ . Find $b$ , $c$ , $d$ and $e$ .
[4 marks]
Answer

Practice

A-Level

C1
Sketch the curve $y = (x + 1) (x - 2) (3 - x) (x + 4)$ , showing the coordinates of the points where it meets the axes.
[4 marks]

C2
Show that the curve $y = (x + 2) (x^{2} - 2 x + 5)$ meets the $x$ -axis at only one point. Sketch the curve, showing where it meets the axes.
[4 marks]
C3
Factorise $(x^{2} + 2 x - 15) (x^{2} - 6 x + 9)$ completely. Hence sketch $y = (x^{2} + 2 x - 15) (x^{2} - 6 x + 9)$ , showing the coordinates of the points where it meets the axes.
[5 marks]

Exam-style questions

Edexcel exam style

D1
$f (x) = (x + 4) (2 x^{2} - 2 x) - 12 x$
(a)
Express $f (x)$ in the form $x (a x^{2} + b x + c)$ , where $a$ , $b$ and $c$ are constants.
[2 marks]
Answer
(b)
Hence factorise $f (x)$ completely.
[2 marks]
Answer
(c)
Sketch the curve $y = f (x)$ , showing the coordinates of the points where it meets the axes.
[3 marks]
(d)
Hence find the set of values of $x$ for which $f (x) > 0$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Sam solves $(x - 4) (x + 1)^{2} \leq 0$ . He writes: "The curve is below the $x$ -axis between its roots, so the answer is $- 1 \leq x \leq 4$ ."
Sam is wrong.
(a)
Give a value of $x$ that is not in Sam's answer but does satisfy $(x - 4) (x + 1)^{2} \leq 0$ , and explain what Sam has missed.
[2 marks]
(b)
Sketch $y = (x - 4) (x + 1)^{2}$ and find, in set notation, the correct set of values of $x$ .
[4 marks]
Answer
Total for D2: 6 marks
D3
The curve $C$ has equation $y = (x - 1) (x^{2} + k x + 16)$ , where $k$ is a constant.
(a)
Find the set of values of $k$ for which $C$ meets the $x$ -axis at exactly one point.
[4 marks]
Answer
(b)
Given that $k = 8$ , sketch $C$ , showing the coordinates of the points where it meets the axes.
[3 marks]
Total for D3: 7 marks
D4
The sketch shows the curve $C$ with equation $y = a (x + p) (x - q)^{2}$ , where $a$ , $p$ and $q$ are constants. $C$ crosses the $x$ -axis at $x = - 2$ , touches it at $x = 3$ , and passes through $(1, - 12)$ .
(a)
Find the values of $a$ , $p$ and $q$ .
[3 marks]
Answer
(b)
Write the equation of $C$ in the form $y = a x^{3} + b x^{2} + c x + d$ .
[2 marks]
Answer
(c)
The line $y = x - 3$ meets $C$ at three points. Using your answer to part (a), find the exact $x$ -coordinates of these points.
[3 marks]
Answer
Total for D4: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The curve $y = (x^{2} - 4) (x^{2} - k)$ , where $k$ is a constant, can meet the $x$ -axis at two, three or four points. For each of these numbers, find the set of values of $k$ that gives it.
[6 marks]
Hint 1 · What to try
The roots are $x = \pm 2$ and the solutions of $x^{2} = k$ . How many solutions can $x^{2} = k$ have?
Hint 2 · The first line
$k < 0$ : none. $k = 0$ : one, $x = 0$ . $k > 0$ : two, $\pm k$ , which are $\pm 2$ when $k = 4$ .
Hint 3 · The full method
Two points: $k < 0$ or $k = 4$ . Three: $k = 0$ . Four: $k > 0$ , $k \neq = 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch a cubic or quartic from its factors.
I can show where a curve touches the axis.
I can find an equation from a sketch.
I can solve an inequality from a sketch.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers