A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 2: Functions and graphs · Lesson 4

Modulus graphs and combined transformations

118 marks

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Date

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Quick questions to start.

A1
From Year 1
The point $(3, - 4)$ lies on the curve $y = f (x)$ . Write down its image on the curve with equation
(a) $y = f (x + 2)$
(b) $y = 2 f (x)$
[2 marks]
Answer
A2
Lesson 1
Solve $∣2 x - 3∣ = 5$ .
[2 marks]
Answer
A3
From Year 1
The curve $y = f (x)$ has a maximum point at $(2, 5)$ . Write down the coordinates of the maximum point of $y = f (3 x)$ .
[2 marks]
Answer
A4
Needed today
$f (x) = x^{2} - 4 x$ . Find the points where $y = f (x)$ meets the $x$ -axis, and the coordinates of its turning point.
[3 marks]
Answer

Example, then your turn

A-Level

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

Example 1

$f (x) = x^{2} - 2 x - 3$ . Sketch the graph of $y = ∣ f (x) ∣$ . Give the coordinates of the points where it meets the axes, and of its maximum point.

Hint Draw

y = f (x)

first. Keep the part on or above the

x

-axis. Reflect the part below the

x

-axis in the

x

-axis.

B1
$f (x) = x^{2} - 6 x + 5$ . Sketch the graph of $y = ∣ f (x) ∣$ . Give the coordinates of the points where it meets the axes, and of its maximum point.
[4 marks]

Example 2

$f (x) = x^{2} - 4 x + 3$ . Sketch the graph of $y = f (∣ x ∣)$ . Give the coordinates of the points where it meets the axes, and of its minimum points.

Hint For

x \geq 0

f (∣ x ∣) = f (x)

. For

x < 0

f (∣ x ∣) = f (- x)

, so the left half is the reflection of the right half in the

y

-axis.

B2
$f (x) = x^{2} - 6 x + 8$ . Sketch the graph of $y = f (∣ x ∣)$ . Give the coordinates of the points where it meets the axes, and of its minimum points.
[4 marks]

Example 3

The curve $y = f (x)$ has a turning point at $(1, 4)$ . Find the coordinates of the turning point on the curve $y = 2 f (x + 3) - 5$ .

Hint The changes inside the bracket act on

x

. The changes outside act on

y

. Find the

x

that makes the bracket equal to

1

B3
The curve $y = f (x)$ has a turning point at $(4, - 6)$ . Find the coordinates of the turning point on the curve $y = 3 f (2 x - 2) + 1$ .
[3 marks]
Answer

Example 4

Sketch the graph of $y = 3∣ x + 1∣ - 6$ . Give the coordinates of its vertex and of the points where it meets the axes.

Hint

y = ∣ x ∣

is translated

1

to the left, stretched by

3

parallel to the

y

-axis, then translated

6

down.

B4
Sketch the graph of $y = 2∣ x - 4∣ - 6$ . Give the coordinates of its vertex and of the points where it meets the axes.
[4 marks]

Example 5

The diagram shows the curve $y = f (x)$ , with a maximum point $A (- 2, 3)$ and a minimum point $B (1, - 2)$ . Give the coordinates of the images of $A$ and $B$ on

(a) $y = ∣ f (x) ∣$

(b) $y = f (∣ x ∣)$

Hint For

∣ f (x) ∣

only the

y

-coordinate of a point below the axis changes sign. For

f (∣ x ∣)

only points with

x \geq 0

are kept, together with their reflections in the

y

-axis.

B5
The curve $y = f (x)$ has a maximum point $P (- 3, 5)$ and a minimum point $Q (2, - 4)$ . It has no other turning points. Give the coordinates of the turning points of
(a) $y = ∣ f (x) ∣$ that come from $P$ and $Q$
(b) $y = f (∣ x ∣)$ that come from $P$ and $Q$
[4 marks]
Answer

Practice

A-Level

C1
$f (x) = x - 3$ . Raj says: "The graphs of $y = ∣ f (x) ∣$ and $y = f (∣ x ∣)$ are the same."
(a) Work out $∣ f (x) ∣$ and $f (∣ x ∣)$ when $x = 0$ , and when $x = - 5$ .
(b) Find the values of $x$ for which $∣ f (x) ∣ = f (∣ x ∣)$ .
[5 marks]
Answer

C2
The point $(4, 1)$ lies on $y = f (x)$ . Ella says: "To get $y = f (2 x + 6)$ , stretch $y = f (x)$ parallel to the $x$ -axis by scale factor $\frac{1}{2}$ , then translate it $6$ to the left."
(a) Find the image of $(4, 1)$ on $y = f (2 x + 6)$ .
(b) Find where Ella's two steps send $(4, 1)$ , and describe the transformation correctly.
[4 marks]
Answer
C3
(a) Sketch the graph of $y = ∣ ln x ∣$ , $x > 0$ . Give the coordinates of the point where it meets the $x$ -axis.
(b) Solve $∣ ln x ∣ = 2$ .
[4 marks]
C4
Sketch the graph of $y = 5 - 2∣ x + 1∣$ . Give the coordinates of its vertex and of the points where it meets the axes, and state the range.
[5 marks]
C5
The curve $y = f (x)$ has turning points $P (- 2, 6)$ and $Q (4, - 3)$ . Find the coordinates of the images of $P$ and $Q$ on
(a) $y = - 2 f (x + 1)$
(b) $y = f (\frac{1}{2} x) - 3$
[4 marks]
Answer
C6
$f (x) = (x - 1) (x - 3) (x + 2)$ .
(a) Write down the roots of $f (∣ x ∣) = 0$ and the $y$ -intercept of $y = f (∣ x ∣)$ .
(b) The minimum value of $f (x)$ for $x \geq 0$ is about $- 4.06$ . How many solutions does $f (∣ x ∣) = 2$ have? Give a reason.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The function $f$ is defined by $f (x) = x^{2} - 6 x$ , $x \in R$ .
(a)
Sketch the graph of $y = ∣ f (x) ∣$ . Give the coordinates of the points where it meets the axes, and of its maximum point.
[3 marks]
(b)
Sketch the graph of $y = f (∣ x ∣)$ . Give the coordinates of the points where it meets the axes, and of its minimum points.
[3 marks]
(c)
Find the coordinates of the turning point of $y = 2 f (x - 1) + 5$ .
[2 marks]
Answer
(d)
State the number of solutions of $∣ f (x) ∣ = k$ when $0 < k < 9$ .
[2 marks]
Answer
Total for D1: 10 marks

D2
Ben says: "For any function $f$ , the graphs of $y = ∣ f (x) ∣$ and $y = f (∣ x ∣)$ are both symmetrical about the $y$ -axis."
Ben is wrong.
(a)
Show that the graph of $y = f (∣ x ∣)$ is always symmetrical about the $y$ -axis.
[2 marks]
(b)
$f (x) = x^{2} - 4 x - 5$ . Show that the graph of $y = ∣ f (x) ∣$ is not symmetrical about the $y$ -axis.
[2 marks]
(c)
For the same $f$ , sketch the graph of $y = f (∣ x ∣)$ . Give the coordinates of the points where it meets the axes, and of its minimum points.
[4 marks]
Total for D2: 8 marks
D3
The curve $y = f (x)$ has a maximum point $P (2, 7)$ and passes through $Q (0, 3)$ .
Describe fully a sequence of transformations that maps $y = f (x)$ onto $y = 4 - 2 f (3 x + 6)$ . Find the coordinates of the images of $P$ and $Q$ , and state whether the image of $P$ is a maximum or a minimum point.
[9 marks]
D4
The function $f$ is defined by $f (x) = 2∣ x + 1∣ - 3$ , $x \in R$ .
(a)
Sketch the graph of $y = f (x)$ . Give the coordinates of its vertex and of the points where it meets the axes.
[4 marks]
(b)
State the range of $f$ and explain why $f$ has no inverse.
[2 marks]
(c)
Give the coordinates of the maximum point of $y = ∣ f (x) ∣$ .
[2 marks]
Answer
(d)
Find the set of values of $k$ for which $∣ f (x) ∣ = k$ has exactly four solutions.
[2 marks]
Answer
Total for D4: 10 marks
D5
The function $f$ is defined by $f (x) = e^{x} - 4$ , $x \in R$ .
(a)
Sketch the graph of $y = ∣ f (x) ∣$ . Give the coordinates of the points where it meets the axes, and the equation of its asymptote.
[3 marks]
(b)
Sketch the graph of $y = f (∣ x ∣)$ . Give the coordinates of the points where it meets the axes.
[3 marks]
(c)
The curve $y = 2 f (x - 1)$ meets the $x$ -axis at $A$ and has the horizontal asymptote $y = c$ . Find the exact $x$ -coordinate of $A$ and the value of $c$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
The diagram shows the curve $y = f (x)$ , where $f (x) = x^{2} + 2 x - 3$ . It meets the $x$ -axis at $(- 3, 0)$ and $(1, 0)$ and has a minimum point at $(- 1, - 4)$ .
(a)
Sketch $y = ∣ f (x) ∣$ . Give the coordinates of its maximum point and of the points where it meets the axes.
[3 marks]
(b)
Give the coordinates of the minimum point of $y = f (∣ x ∣)$ and the points where it meets the $x$ -axis.
[2 marks]
Answer
(c)
The curve $y = a f (x + b)$ has a minimum point at $(2, - 12)$ . Find the values of $a$ and $b$ .
[3 marks]
Answer
Total for D6: 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 $C$ has equation $y = ∣ x^{2} - 6∣ x ∣ + 5∣$ . Find the set of values of $k$ for which the equation $∣ x^{2} - 6∣ x ∣ + 5∣ = k$ has exactly eight solutions, and the value of $k$ for which it has exactly three solutions.
[9 marks]
Hint 1 · What to try
Start with $y = x^{2} - 6 x + 5 = (x - 1) (x - 5)$ . Sketch $y = f (∣ x ∣)$ first, then take the modulus of that.
Hint 2 · The first line
For $x \geq 0$ the curve $C$ falls from $5$ at $x = 0$ to $0$ at $x = 1$ , rises to a maximum $4$ at $x = 3$ , falls to $0$ at $x = 5$ , then rises. The left half is the reflection in the $y$ -axis.
Hint 3 · The full method
Count where a horizontal line $y = k$ meets $C$ . Look at $k$ below $4$ , between $4$ and $5$ , and equal to $5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch y = \|f(x)\| by reflecting the parts of y = f(x) below the x-axis.
I can sketch y = f(\|x\|) by keeping the part of y = f(x) for x ≥ 0 and reflecting it in the y-axis.
I can find where a point moves under a combined transformation y = af(bx + c) + d.
I can describe a combined transformation in the right order.

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