KS3 Maths · Year 8 · Chapter 11: Solving equations graphically · Lesson 4

Solving cubic equations by drawing graphs

Year 8About 75 minutesNo calculator46 marks

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Quick questions to start.

A1
Last lesson
The lowest point of $y = x^{2} - 6 x + 1$ is $(3, - 8)$ . How many solutions does $x^{2} - 6 x + 1 = - 8$ have?
[1 mark]
Answer
A2
Needed today
Work out the value of $x^{3} - 5 x$ when $x = - 2$ .
[1 mark]
Answer
A3
From chapter 9
Write $3.2 \times 1 0^{- 2}$ as an ordinary number.
[1 mark]
Answer
A4
Year 7
Solve $2 x + 7 = 1$ .
[1 mark]
Answer

Example, then your turn

Year 8

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

Example 1

Complete the table of values for $y = x^{3} + 2 x^{2}$ . Use it to solve $x^{3} + 2 x^{2} = 0$ .

Hint Cube

x

(keep the sign), then add

2 x^{2}

. The solutions are where

y = 0

B1
Complete the table of values for $y = x^{3} - x^{2}$ . Use it to solve $x^{3} - x^{2} = 0$ .
[3 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Example 2

The graph of $y = x^{3} - 7 x$ is drawn. (a) Use it to solve $x^{3} - 7 x = 6$ . (b) How many solutions do $x^{3} - 7 x = 10$ and $x^{3} - 7 x = 0$ have?

Hint Draw each line

y = 6

y = 10

y = 0

. Count where each one meets the curve.

B2
The graph of $y = x^{3} - 13 x$ is drawn. (a) Use it to solve $x^{3} - 13 x = 12$ . (b) How many solutions does $x^{3} - 13 x = 20$ have?
[3 marks]
Answer

Example 3

The graph of $y = x^{3} - 3 x^{2}$ is drawn. Use it to solve $x^{3} - 3 x^{2} + 2 = 0$ . Give your answers to $1$ decimal place.

Hint Take

2

from both sides to get

x^{3} - 3 x^{2} = - 2

. Draw the line

y = - 2

. Read off each crossing; two of them are between gridlines.

B3
The graph of $y = x^{3} + 3 x^{2}$ is drawn. Draw a suitable line and use it to solve $x^{3} + 3 x^{2} - 2 = 0$ . Give your answers to $1$ decimal place.
[3 marks]
Answer $x = \dots\dots$ , $x = \dots\dots$ , $x = \dots\dots$

Practice

Year 8

C1
The sketch shows a cubic curve. Its bends are at $(- 1, 5)$ and $(2, - 3)$ . How many times does the line $y = k$ meet the curve when (a) $k = 0$ , (b) $k = 5$ , (c) $k = 8$ ?
[3 marks]
Not drawn accurately
Answer
C2
The curve $y = x^{3} - 2 x^{2}$ and the line $y = 9$ are drawn. Which equation do their crossing points solve? Write it in the form $x^{3} + a x^{2} + c = 0$ , and write down its solution.
[2 marks]
Answer

C3
Ben says, "Some cubic equations have no solution." Use the shape of a cubic graph to explain why Ben is wrong.
[2 marks]
C4
The graph of $y = x^{3} - 12 x$ is drawn. For which values of $k$ does $x^{3} - 12 x = k$ have three solutions?
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Leo uses the graph of $y = x^{3} - 3 x^{2} + 4$ to solve $x^{3} - 3 x^{2} + 4 = 20$ . He says, "It is a cubic, so it must have three solutions."
Explain why Leo is wrong. Use the graph to solve the equation.
[4 marks]
Answer $x = \dots\dots$

D2
A box has a base $x$ cm by $2 x$ cm. Its height is $1$ cm more than $x$ . Its volume is $V$ cm³.
(a)
Show that $V = 2 x^{3} + 2 x^{2}$ .
[1 mark]
(b)
Work out $V$ when $x = 1$ , $2$ and $3$ . Plot the points on the grid and draw the graph of $V = 2 x^{3} + 2 x^{2}$ through them and $(0, 0)$ .
[2 marks]
(c)
The volume of the box is $60$ cm³. Use your graph to find $x$ . Give your answer to $1$ decimal place.
[2 marks]
Answer
Total for D2: 5 marks
D3
The graph of $y = x^{3} - x^{2} - 4 x + 4$ is drawn on the grid.
(a)
Use the graph to solve $x^{3} - x^{2} - 4 x + 4 = 0$ .
[2 marks]
Answer $x = \dots\dots$ , $x = \dots\dots$ , $x = \dots\dots$
(b)
By drawing a suitable line, solve $x^{3} - x^{2} - 4 x + 3 = 0$ . Give your answers to $1$ decimal place.
[2 marks]
Answer $x = \dots\dots$ , $x = \dots\dots$ , $x = \dots\dots$
(c)
How many solutions does $x^{3} - x^{2} - 4 x + 4 = - 3$ have? Explain how you know from the graph.
[2 marks]
Total for D3: 6 marks
D4
The graph of $y = x^{3} - 3 x$ and the line $y = x$ are drawn on the grid.
(a)
Write down the coordinates of the three points where the line meets the curve.
[2 marks]
Answer
(b)
How many solutions does $x^{3} - 3 x = 2$ have? Explain how you know from the graph.
[2 marks]
Total for D4: 4 marks

Extension

Stretch

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

E1
The curve $y = x^{3} - 3 x^{2} - 9 x$ has its two bends at $x = - 1$ and $x = 3$ . The equation $x^{3} - 3 x^{2} - 9 x + k = 0$ has exactly two solutions. Find the two possible values of $k$ .
[5 marks]
Hint 1 · What to try
Write it as $x^{3} - 3 x^{2} - 9 x = - k$ . Exactly two solutions means the line $y = - k$ just touches a bend.
Hint 2 · The first line
Work out the $y$ -value of each bend: put $x = - 1$ and $x = 3$ into $x^{3} - 3 x^{2} - 9 x$ .
Hint 3 · The full method
The bends are $(- 1, 5)$ and $(3, - 27)$ . So $- k = 5$ or $- k = - 27$ .
Answer $k = \dots\dots$ or $k = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can complete a table of values for a cubic and read solutions where $y = 0$ .
I can draw a line $y = k$ on a cubic graph and read the solutions.
I can say whether a cubic equation has three, two or one solution.

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