Transformations Flashcards

All 29 cards in this deck

In a translation vector $\binom{a}{b}$, what do the top and bottom numbers tell you?

e.g. $\binom{3}{-2}$

Top = movement parallel to the $x$-axis (positive = right), bottom = movement parallel to the $y$-axis (positive = up).

e.g. 3 right and 2 down.

What must you state to describe fully a translation?

Translation, and the column vector, e.g. translation by vector $\binom{-5}{6}$ (not the coordinate $(-5,6)$).

True or false? A translation can change the size or orientation of a shape.

False. The image is congruent to the object and has the same orientation; only its position changes.

What are the steps to translate a shape on a grid by a given vector?

e.g. translate by $\binom{-1}{-2}$

1. Pick a vertex and count the vector across then up/down: 1 left, 2 down.
2. Plot the image of that vertex, then repeat for every vertex.
3. Join the image vertices in the same order and label the image.

What happens to the centre and radius of a circle when the circle is translated by a vector?

e.g. $x^2+y^2=16$ translated by $\binom{3}{0}$

The centre moves by the vector; the radius is unchanged.

e.g. centre $(3,0)$, radius $4$.

What single transformation maps the image back onto the object after a translation by vector $\binom{a}{b}$?

e.g. translation by $\binom{4}{-1}$

Translation by $\binom{-a}{-b}$.

e.g. translation by $\binom{-4}{1}$.

On a coordinate grid, what do the mirror lines $x=3$ and $y=-2$ look like?

$x=3$ is a vertical line through $(3,0)$; $y=-2$ is a horizontal line through $(0,-2)$.

On a coordinate grid, what do the mirror lines $y=x$ and $y=-x$ look like?

Both are diagonal lines through the origin: $y=x$ has gradient $1$ (through $(1,1)$), $y=-x$ has gradient $-1$ (through $(1,-1)$).

What must you state to describe fully a reflection?

Reflection, and the equation of the mirror line, e.g. reflection in the line $x=-1$.

What are the steps to reflect a shape in a given mirror line?

e.g. reflect in the line $y=x$

1. Measure the perpendicular distance from a vertex to the mirror line.
2. Plot the image vertex the same distance on the other side, on that perpendicular.
3. Repeat for all vertices and join them; for $y=x$ each point $(x,y)$ lands at $(y,x)$.

True or false? Any point lying on the mirror line is invariant under the reflection.

True. Points on the mirror line map onto themselves.

A graph sketched on axes is reflected in the $x$-axis. What happens to the coordinates of each point?

$(x,y)\to(x,-y)$ — the $x$-coordinate stays the same and the $y$-coordinate changes sign.

What must you state to describe fully a rotation?

Rotation, the angle, the direction (clockwise or anticlockwise) and the centre of rotation.

True or false? A rotation of $180^\circ$ must have its direction stated to be fully described.

False. Clockwise and anticlockwise $180^\circ$ give the same image, so only the angle and centre are needed.

A rotation of $90^\circ$ anticlockwise about a point is the same as which other rotation about that point?

A rotation of $270^\circ$ clockwise.

What are the steps to rotate a shape about a given centre?

e.g. rotate $90^\circ$ clockwise about the origin

1. Trace the shape and mark the centre on the tracing paper.
2. Hold a pencil point on the centre and turn the paper $90^\circ$ clockwise.
3. Mark the new vertex positions, join them and label the image.

What is the coordinate rule for a rotation of $180^\circ$ about the origin?

$(x,y)\to(-x,-y)$.

Which point is always invariant under a rotation?

The centre of rotation.

What must you state to describe fully an enlargement?

Enlargement, the scale factor, and the centre of enlargement, e.g. enlargement scale factor $3$, centre $(0,2)$.

What are the steps to enlarge a shape by a positive scale factor from a given centre?

e.g. scale factor $2$, centre $(0,1)$

1. Count the steps across and up from the centre to a vertex.
2. Multiply both counts by the scale factor ($\times 2$) and count again from the centre to plot the image vertex.
3. Repeat for every vertex, then join them.

What is the effect of enlarging a shape by a fractional scale factor between $0$ and $1$?

e.g. scale factor $\frac{1}{3}$

The image is smaller than the object and closer to the centre of enlargement, with the same orientation; each length is multiplied by $\frac{1}{3}$.

What is the effect of enlarging a shape by a negative scale factor?

e.g. scale factor $-2$

The image appears on the opposite side of the centre of enlargement and is turned upside down (rotated $180^\circ$), with lengths multiplied by $2$.

True or false? An enlargement of scale factor $-1$ about a point gives the same image as a rotation of $180^\circ$ about that point.

True. Both are accepted as descriptions of that transformation.

True or false? An enlargement changes the angles inside a shape.

False. Angles are unchanged (the shapes are similar); only the lengths are multiplied by the scale factor.

What does it mean to say a point is invariant under a transformation?

The point maps onto itself — it stays in exactly the same position.

What single transformation is equivalent to two reflections in parallel mirror lines?

e.g. reflect in $x=2$, then in $x=6$

A translation perpendicular to the lines, of twice the distance between them.

e.g. translation by vector $\binom{8}{0}$.

What single transformation is equivalent to two reflections in perpendicular mirror lines?

e.g. reflect in the $x$-axis, then in the $y$-axis

A rotation of $180^\circ$ about the point where the two lines cross.

e.g. rotation of $180^\circ$ about $(0,0)$.

True or false? Doing two transformations in the opposite order always gives the same final image.

False. Order usually matters, e.g. reflecting in the $x$-axis then in $y=x$ gives a $90^\circ$ rotation one way, the other order gives the $90^\circ$ rotation the other way.

When asked to describe fully the single transformation equivalent to a combination, how many transformations may your answer contain?

Exactly one — no marks are given if more than one transformation is described.