Combined Transformations 1

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Combined Transformations 1 — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionApplying Successive TransformationsAssessment Practice

5 marks~8 minCriterion A

The diagram shows a coordinate grid with point

P(2, 3)

.

Point

P

is first reflected in the

y

-axis to give image

P'

, then

P'

is translated by the vector

\begin{pmatrix}4 \\ -1\end{pmatrix}

to give image

P''

Write down the coordinates of

P'

. [1]

Find the coordinates of

P''

. [1]

A third transformation maps

P''

back to the original point

P(2, 3)

. The transformation is a reflection in the line

y = c

, where

c

is a constant.

Find the value of

c

and justify whether this mirror line lies closer to

P''

or to

P

. [3]

Solutions

2QuestionApplying Successive TransformationsAssessment Practice

6 marks~9 minCriterion C

The diagram shows the graph of

f(x) = x^2 - 4x + 3

for

0 \le x \le 5

Write down the coordinates of the

y

-intercept of

f(x)

. [1]

Find the coordinates of the vertex of

f(x)

by completing the square, giving your answer in exact form. [2]

Justify the set of values of

k

for which

f(x) = k

has exactly two distinct solutions in the domain

0 \le x \le 5

. [3]

Solutions

3QuestionApplying Successive TransformationsAssessment Practice

6 marks~9 minCriterion D

A delivery drone operates inside a large warehouse. Its position is tracked on a coordinate grid where each unit represents one metre. The drone starts at point

A(1, 2)

.

The warehouse manager uses the following model to predict the drone's position after

n

complete delivery runs. After each run, the drone moves 3 metres east (positive

x

-direction) and 2 metres north (positive

y

-direction) from its previous position.

Write down the coordinates of the drone after run 1 and after run 2. [1]

The drone must stay within the rectangular boundary

0 \le x \le 16

and

0 \le y \le 12

. Find the maximum number of complete delivery runs the drone can make before it first exits this boundary. Show your reasoning clearly. [3]

The manager claims the model predicts the drone will always remain inside the boundary provided the number of runs does not exceed the value found in part (b). Identify one assumption of this translation model and advise the manager whether this claim can be relied upon in a real warehouse, justifying your answer. [2]

Solutions

4QuestionApplying Successive TransformationsAssessment Practice

7 marks~11 minCriterion B

The diagram shows the first four points of a pattern. Starting from

P_0(1, 2)

, each new point is obtained by applying the same translation to the previous point. The translation moves each point 3 units in the positive

x

-direction and 2 units in the positive

y

-direction.

Write down the coordinates of

P_1

P_2

P_3

, and

P_4

. [2]

Find a general expression for the coordinates of

P_n

in terms of

n

, where

n

is a non-negative integer. [2]

Justify why both coordinate expressions are linear in

n

, and verify your expression by substituting

n = 3

. [3]

Solutions

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