A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 4: Graphs and transformations · Lesson 2

Reciprocal graphs and points of intersection

70 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Solve $4 x^{2} + 5 x - 6 = 0$ .
[2 marks]
Answer
A2
Last lesson
Write down the coordinates of the points where $y = (x - 3) (x + 2)^{2}$ meets the axes.
[2 marks]
Answer
A3
From chapter 2
Find the set of values of $k$ for which $x^{2} + 8 x + k = 0$ has no real roots.
[2 marks]
Answer
A4
Needed today
Find the coordinates of the points where $y = x^{2} - 3$ meets $y = 2 x$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Sketch $y = - \frac{3}{x}$ , stating the equations of its asymptotes.

Hint

y = \frac{k}{x}

with

k < 0

: when

x > 0

y < 0

, and when

x < 0

y > 0

B1
Sketch $y = \frac{5}{x ^{2}}$ , stating the equations of its asymptotes. Write down the set of values that $y$ takes.
[3 marks]

Example 2

On the same axes sketch $y = \frac{2}{x}$ and $y = \frac{8}{x}$ . Find the set of values of $x$ for which $\frac{8}{x} > \frac{2}{x}$ .

Hint For

x > 0

the bigger numerator gives the bigger value. For

x < 0

both are negative.

B2
On the same axes sketch $y = - \frac{2}{x ^{2}}$ and $y = - \frac{7}{x ^{2}}$ . Find the set of values of $x$ for which $- \frac{7}{x ^{2}} < - \frac{2}{x ^{2}}$ .
[4 marks]
Answer

Example 3

On the same axes sketch $y = x^{2} (x + 3)$ and $y = - \frac{2}{x ^{2}}$ . Hence state, with a reason, the number of real solutions of $x^{4} (x + 3) + 2 = 0$ .

Hint Divide

x^{4} (x + 3) + 2 = 0

x^{2}

: it says

x^{2} (x + 3) = - \frac{2}{x ^{2}}

, so count where the curves meet.

B3
On the same axes sketch $y = (x + 2)^{2} (x - 1)$ and $y = \frac{4}{x}$ , showing where the cubic meets the axes. Hence state, with a reason, the number of real solutions of $x (x + 2)^{2} (x - 1) = 4$ .
[4 marks]
Answer

Example 4

Find the coordinates of the points where $y = \frac{10}{x}$ meets $y = 2 x + 1$ .

Hint Set

\frac{10}{x} = 2 x + 1

and multiply by

x

B4
Find the coordinates of the points where $y = \frac{15}{x}$ meets $y = 3 x - 4$ .
[5 marks]
Answer

Practice

A-Level

C1
On the same axes sketch $y = \frac{3}{x}$ and $y = x^{2} - 4$ . Hence state, giving a reason, the number of real solutions of $x^{3} - 4 x - 3 = 0$ .
[4 marks]
Answer

C2
$a$ and $b$ are positive constants. On the same axes sketch $y = - \frac{a}{x}$ and $y = x (x + b)^{2}$ , showing where each curve meets the axes. Explain why $x^{2} (x + b)^{2} + a = 0$ has no real solutions.
[4 marks]
C3
On the same axes sketch $y = x^{3} + x^{2} - 9 x$ and $y = 3 x$ . Find the coordinates of the points where they meet.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = \frac{9}{x ^{2}}$ and the line $L$ has equation $y = 4 x + 13$ .
(a)
On the same axes sketch $C$ and $L$ , showing where $L$ meets the axes.
[3 marks]
(b)
Using your sketch, state the number of points where $C$ and $L$ meet.
[1 mark]
Answer
(c)
Show that the $x$ -coordinates of these points satisfy $(x + 1) (x + 3) (4 x - 3) = 0$ .
[2 marks]
(d)
Hence find the exact coordinates of the points where $C$ and $L$ meet.
[3 marks]
Answer
Total for D1: 9 marks

D2
Ravi sketches $y = x (x - 5)$ and $y = - \frac{4}{x}$ . He says: "For $x < 0$ the curves never meet, so $x (x - 5) = - \frac{4}{x}$ has exactly one real solution."
Ravi is wrong.
(a)
Show that the equation can be written as $x^{3} - 5 x^{2} + 4 = 0$ , and verify that $x = 1$ is a solution.
[2 marks]
(b)
Hence find all the solutions exactly, and say what Ravi got wrong.
[3 marks]
Answer
Total for D2: 5 marks
D3
The curve $C$ has equation $y = \frac{4}{x}$ and the line $L$ has equation $y = k - x$ , where $k$ is a constant. Find the set of values of $k$ for which $L$ meets $C$ at two distinct points.
[5 marks]
Answer
D4
The curve $C$ has equation $y = \frac{12}{x}$ and the curve $D$ has equation $y = x^{2} + x$ .
(a)
On the same axes sketch $C$ and $D$ , showing where $D$ meets the $x$ -axis.
[2 marks]
(b)
Show that the $x$ -coordinates of the points where $C$ and $D$ meet satisfy $x^{3} + x^{2} - 12 = 0$ , and that $x = 2$ is one of them.
[2 marks]
(c)
Hence show that $C$ and $D$ meet at exactly one point, and find its coordinates.
[4 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 $y = \frac{k}{x ^{2}}$ , where $k$ is a non-zero constant, and the line $y = 3 - x$ can meet at one, two or three points. For each of these numbers, find the set of values of $k$ that gives it.
[6 marks]
Hint 1 · What to try
They meet where $k = x^{2} (3 - x)$ , $x \neq = 0$ . So count where the line $y = k$ meets $y = x^{2} (3 - x)$ .
Hint 2 · The first line
$y = k$ touches the cubic where $x^{3} - 3 x^{2} + k$ has a repeated root. Write it as $(x - a)^{2} (x - b)$ and compare coefficients.
Hint 3 · The full method
$2 a + b = 3$ and $a^{2} + 2 ab = 0$ , so $a = 2$ , $b = - 1$ and $k = 4$ : the cubic touches the axis at $0$ and has its top at $(2, 4)$ . For $0 < k < 4$ , $y = k$ meets it once for $x < 0$ and twice for $0 < x < 3$ : three points. For $k = 4$ : at $x = 2$ and once for $x < 0$ , two points. For $k > 4$ only $x < 0$ , and for $k < 0$ only $x > 3$ : one point.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch y = k/x and y = k/x².
I can count solutions from a sketch.
I can find where two curves meet.
I can find when a line meets a curve twice.

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