Combined Transformations 1

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Combined Transformations 1 — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionIdentifying Single Equivalent TransformationConcept Practice

4 marks~6 minCriterion A

The diagram shows a coordinate grid with point

P(2, 3)

, its image

P'

after reflection in the

x

-axis, and its image

P''

after a further reflection in the

y

-axis.

Write down the coordinates of

P'

, the image of

P(2, 3)

after reflection in the

x

-axis. [1]

Write down the coordinates of

P''

, the image of

P'

after reflection in the

y

-axis. [1]

Justify, using correct transformation notation and by referring to the coordinates of

P

and

P''

, that the combined effect of the two reflections is equivalent to a single rotation of

180°

about the origin. [2]

Solutions

2QuestionApplying Successive TransformationsAssessment Practice

5 marks~8 minCriterion C

The diagram shows a coordinate grid with point

P(2, 3)

.

Point

P

is first rotated

90^\circ

anticlockwise about the origin to give image

P'

, then

P'

is translated by

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

to give final image

P''

Write down the coordinates of

P'

. [1]

Find the coordinates of

P''

. Show your working clearly. [2]

A classmate claims that applying the translation first and then the rotation gives the same final image

P''

. Justify whether this claim is correct, showing all working. [2]

Solutions

3QuestionUsing Matrices for Simple Transformations IntroductoryAssessment Practice

9 marks~14 minCriterion D

A game developer is designing a 2D platformer. The hero sprite starts as a unit square with vertices at

A(0,0)

B(1,0)

C(1,1)

and

D(0,1)

. To fit a widescreen display, the developer applies a stretch mapping each point

(x, y)

(3x, 2y)

.

The diagram shows the original sprite

ABCD

and the image

A'B'C'D'

after the stretch.

Write down the coordinates of

A'

B'

C'

and

D'

. [1]

The developer changes the stretch so that the horizontal factor is

k

and the vertical factor is

(k-1)

, where

k > 1

. The stretched sprite must fit exactly inside a screen region of area 30 square units. Find the value of

k

, giving your answer to 3 significant figures. [3]

The developer tests three stretch settings and records the predicted sprite area alongside the actual rendered area:

k

: 3, 4, 6

Predicted area (

k(k-1)

): 6, 12, 30

Actual area: 5, 11, 28

The percentage error is defined as

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Calculate the percentage error at each value of

k

, giving your answers to 1 decimal place. [3]

Using your results from part (c), justify whether the developer should use the model

\text{Area} = k(k-1)

to predict rendered sprite areas in production. [2]

Solutions

4QuestionApplying Successive TransformationsAssessment Practice

11 marks~17 minCriterion B

The diagram shows the first three figures of a pattern made from unit squares arranged in an L-shape.

Figure 1 has 3 unit squares, Figure 2 has 8 unit squares, and Figure 3 has 15 unit squares.

Show that the number of unit squares in Figure 5 is 35 and find the number of unit squares in Figure 6. [3]

A student claims the number of unit squares

S

in Figure

n

is given by

S = n^2 + 2n.

Show that this formula is consistent with Figures 1, 2, and 3, and find the figure number that first contains more than 120 unit squares. [4]

Justify why the formula

S = n^2 + 2n

must contain an

n^2

term by referring to the structure of the figures, and prove algebraically that the difference

S_{n+1} - S_n

is always an odd number. [4]

Solutions

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