8th Grade · 8.G.A.3

Dilations and transformations in 8th grade

Dilations and transformations are the ways a shape can move or change size on a grid. In 8th grade, your child describes each move with coordinates, the pair of numbers $(x, y)$ that gives the position of a point. Slides, flips and turns keep the size of a shape, while a dilation makes it larger or smaller.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

What it means, in plain words

Transformations on the coordinate plane follow simple rules. A translation (a slide) adds the same amounts to every point, or subtracts them. Moving 3 units right and 2 units up turns $(x, y)$ into $(x + 3, y + 2)$ . A reflection (a flip) across the y-axis changes the sign of $x$ . Across the x-axis it changes the sign of $y$ , and across the line $y = x$ the two coordinates swap places. A rotation (a turn) of $90^\circ$ counterclockwise about the origin turns $(x, y)$ into $(-y, x)$ .

Dilations change the size. A dilation centered at the origin multiplies both coordinates by the same number, called the scale factor. With a scale factor of 2, $(3, 5)$ goes to $(6, 10)$ and every length doubles. With a scale factor of $\frac{1}{2}$ , $(4, 6)$ goes to $(2, 3)$ and every length is halved. The angles stay the same, so the image is a larger or smaller copy of the original shape.

Most work with dilations and transformations in 8th grade means applying one of these rules to each corner of a shape and plotting the new corners. Your child then compares the two shapes. After a slide, flip or turn every length is the same. After a dilation every length is multiplied by the scale factor.

Your child should be able to:

Write the new coordinates of a point after a translation, a reflection or a rotation
Dilate a shape from the origin by multiplying both coordinates of each corner by the scale factor
Say whether a scale factor makes a shape larger (more than 1) or smaller (between 0 and 1)
Explain which moves keep the size of a shape and which change it

Example

Triangle $ABC$ has corners $A(1, 2)$ , $B(3, 2)$ and $C(1, 4)$ . Dilate it from the origin with a scale factor of 3. Where are the new corners?

Multiply both coordinates of each corner by 3.
$A(1, 2)$ goes to $A'(3, 6)$ , $B(3, 2)$ goes to $B'(9, 6)$ and $C(1, 4)$ goes to $C'(3, 12)$ .
Side $AB$ was 2 units long. Side $A'B'$ is 6 units long, three times as long.
The new triangle has corners at $(3, 6)$ , $(9, 6)$ and $(3, 12)$ . It has the same shape and is three times as large.

Example

The point $(4, 1)$ is rotated $90^\circ$ counterclockwise about the origin. What is its image?

The rule for a $90^\circ$ counterclockwise turn about the origin is $(x, y) \to (-y, x)$ .
Here $x = 4$ and $y = 1$ , so the new x-coordinate is $-1$ and the new y-coordinate is 4.
The image is the point $(-1, 4)$ , a quarter turn from where the point started.

Where children usually get stuck

To dilate $(3, 5)$ by a scale factor of 2, your child writes $(6, 5)$ .
Only the first coordinate gets multiplied. The dilation is pictured as a stretch in one direction instead of a copy that grows in every direction.
What helps. Plot the origin, $(3, 5)$ and the answer. Ask whether the new point is twice as far from the origin along the same line. For $(6, 10)$ it is, and for $(6, 5)$ it is not.
Your child adds the scale factor instead of multiplying, so $(3, 5)$ with a scale factor of 2 becomes $(5, 7)$ .
Translations work by adding, and the two kinds of rule get mixed up when they appear on the same worksheet.
What helps. Before any numbers, ask what the move does. Adding slides the shape to a new place. Multiplying makes it larger or smaller.
For a $90^\circ$ counterclockwise turn of $(4, 1)$ , your child writes $(1, -4)$ .
The rotation rules are memorized as patterns of swapped numbers and signs, and a memorized pattern is easy to reverse. $(1, -4)$ is the answer for a clockwise turn.
What helps. Plot the point, hold a pencil tip on the origin and turn the paper a quarter turn to the left. Compare where the point ends up with the written answer.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Each row shows a point and a claimed image after reflection across the line y = x. For each row, choose whether the claim is Correct or Incorrect.
Point Claimed image Is the claim correct?
(2, 5) (5, 2)
(3, 1) (3, -1)
(4, 6) (-4, 6)
Show the answer and how to solve it
Choose Correct for the 1st row and Incorrect for the 2nd and 3rd rows, because reflecting across y = x changes (x, y) to (y, x).
1. 1For a reflection across y = x, the coordinates switch places.
2. 2First row: (2, 5) becomes (5, 2). The claimed image matches, so this claim is Correct.
3. 3Second row: (3, 1) should become (1, 3), not (3, -1). The claimed image negated the y-coordinate instead of switching coordinates — this is an x-axis reflection mistake, so the claim is Incorrect.
4. 4Third row: (4, 6) should become (6, 4), not (-4, 6). The claimed image negated the x-coordinate instead of switching coordinates — this is a y-axis reflection mistake, so the claim is Incorrect.
Common mistake. A common mistake is reflecting across an axis instead of across y = x. Negating one coordinate gives a reflection over the x-axis or y-axis, not over the line y = x.
The idea underneath. A reflection across y = x trades a point's horizontal and vertical positions, so the x-coordinate and y-coordinate switch places.
2On level
Point P is dilated with center at the origin using the scale factor shown below. Which coordinate pair is the image of P?
$\displaystyle P(-3, 5),\quad k = 2$
A
$\displaystyle (-6, 10)$
B
$\displaystyle (-1, 7)$
C
$\displaystyle (-6, 5)$
D
$\displaystyle (-3, 10)$
Show the answer and how to solve it
The answer is (-6, 10) because a dilation from the origin with scale factor 2 multiplies both coordinates of P(-3, 5) by 2.
1. 1Look at the original point P(-3, 5) and the scale factor 2.
2. 2A dilation from the origin with scale factor 2 multiplies both coordinates by 2.
3. 3Multiply each coordinate to get the image.
4. 4So the image of P is (-6, 10).
Common mistake. A common mistake is multiplying only one coordinate. In a dilation from the origin, both the x- and y-coordinates must be multiplied by the scale factor.
The idea underneath. A dilation centered at the origin scales the whole position vector, so each coordinate is multiplied by the same positive factor and the point stays in the same quadrant.
3On level
A dilation centered at the origin has the scale factor shown. Which rule matches it?
$\displaystyle k = \frac{1}{2}$
A
$\displaystyle \left(x, y\right) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)$
B
$\displaystyle \left(x, y\right) \rightarrow \left(2x, 2y\right)$
C
$\displaystyle \left(x, y\right) \rightarrow \left(\frac{1}{2}x, y\right)$
D
$\displaystyle \left(x, y\right) \rightarrow \left(-\frac{1}{2}x, -\frac{1}{2}y\right)$
Show the answer and how to solve it
The correct rule is $(x, y) \to \left(\frac{1}{2}x, \frac{1}{2}y\right)$ because a dilation with scale factor $\frac{1}{2}$ multiplies both $x$ and $y$ by $\frac{1}{2}$ .
1. 1The scale factor is $k = \frac{1}{2}$ .
2. 2For a dilation centered at the origin, both coordinates are multiplied by the same scale factor.
3. 3Since $k = \frac{1}{2}$ , both $x$ and $y$ are multiplied by $\frac{1}{2}$ .
4. 4The closest wrong choice uses 2 instead of $\frac{1}{2}$ — that is the reciprocal and describes an enlargement, not a reduction. Another choice only scales $x$ , and another incorrectly flips the signs of both coordinates.
Common mistake. A common mistake is choosing $(2x, 2y)$ . That uses the reciprocal 2 and describes an enlargement instead of the reduction shown by $k = \frac{1}{2}$ .
The idea underneath. For a dilation centered at the origin, the same scale factor must be applied to both coordinates; when $0 < k < 1$ , the image gets smaller because both coordinates are reduced.
4Stretch
Reflect the blue quadrilateral across the y-axis. Which graph shows the reflected image?
A
B
C
D
Show the answer and how to solve it
The correct graph is the one with the red quadrilateral at (5, 2), (3, 5), (0, 3), and (2, 0), because reflecting across the y-axis negates each x-coordinate while the vertex on the y-axis stays fixed.
1. 1Look at the original quadrilateral. Its vertices are (-5, 2), (-3, 5), (0, 3), and (-2, 0).
2. 2A reflection across the y-axis changes each x-coordinate to its opposite and keeps each y-coordinate the same: (x, y) → (-x, y).
3. 3The vertex (0, 3) lies on the y-axis, so it stays fixed after the reflection. The other three vertices move to the right side of the y-axis at the same horizontal distances. The reflected image has vertices (5, 2), (3, 5), (0, 3), and (2, 0).
4. 4The most tempting wrong choice leaves the vertex (-2, 0) in place. Because (-2, 0) is not on the y-axis, it must also be reflected to (2, 0). Only a vertex whose x-coordinate is 0 stays fixed under a y-axis reflection.
Common mistake. A common mistake is leaving the vertex (-2, 0) unchanged. Every vertex whose x-coordinate is not 0 must be moved; only (0, 3) stays fixed because it sits on the y-axis.
The idea underneath. A reflection keeps each point the same distance from the mirror line, so across the y-axis only the horizontal position changes sign while the vertical position stays the same.

Point	Claimed image	Is the claim correct?
(2, 5)	(5, 2)
(3, 1)	(3, -1)
(4, 6)	(-4, 6)