Enlarging with negative scale factors: worked examples and method

Higher · Grade 6 · Non-calculator

This page sets out the method for enlarging with negative scale factors in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Enlarging with negative scale factors topic page.

Key facts

A negative scale factor puts the image on the opposite side of the centre and rotates it by $18 0^{\circ}$ .
Scale factor $- 1$ is the same as a rotation of $18 0^{\circ}$ about the centre.
Image vertex $=$ centre $+ k \times$ (vector from centre to original vertex).
Lengths in the image are $∣ k ∣$ times the original lengths.
A full description needs three things: enlargement, scale factor (with the minus sign), centre.

The method, step by step

Method: Vector from the centre

When you are asked to draw an enlargement with a negative scale factor, work vertex by vertex from the centre.

Mark the centre of enlargement clearly.
For one vertex, count the horizontal and vertical steps from the centre to the vertex.
Multiply both steps by the scale factor — the negative sign reverses both directions.
Count that many steps from the centre to plot the image vertex.
Repeat for every vertex and join them up; the image should look like the original turned upside down.

Method: Describe the transformation

When two shapes are drawn and the image is upside down and on the other side of a point, suspect a negative enlargement.

Join each vertex to its image with a straight line; the lines cross at the centre.
Compare a pair of corresponding lengths to find the size of the scale factor.
Because the image is inverted through the centre, make the scale factor negative.
Write: 'Enlargement, scale factor $k$ , centre $(a, b)$ .'

How the lesson explains it

Through the origin

In an enlargement with centre (0, 0), the image coordinates are the object coordinates multiplied by the scale factor. A negative scale factor puts the image on the opposite side of the centre.

Find the factor

If a map sends (3, 1) to (-6, -2), the scale factor is -2.

Distance

The point (3, 1) is √10 from the origin. After scale factor -2 the image is 2 times as far away, so its distance from the origin is 2√10. The image is on the opposite side of the origin, so the gap between the point and its image is the two distances added.

Quadrant

A first-quadrant point and a negative scale factor land in the third quadrant.

Same factor

Both coordinates use the same scale factor. A coordinate of 0 stays 0.

Do not add

Enlargement multiplies. It does not add the scale factor to the coordinates.

What the minus sign does

Ignore the sign for a moment and look at the size. Scale factor -3 is three times as far from the centre as the object. Scale factor -1/2 is half as far, still on the opposite side, so the image is smaller. The sign does not change the size. It only decides the side of the centre. A positive enlargement of scale factor 3 and a negative enlargement of scale factor -3 give shapes of the same size, but they sit on opposite sides of the centre and one is a half turn of the other.

Picture a point to the right of the centre. After a negative enlargement it is to the left. A point above the centre finishes below it. Both coordinates, measured from the centre, change sign and then stretch by |k|. If the centre is the origin, you can multiply the coordinates by k directly. If the centre is anywhere else, you must measure from the centre first. Multiplying the raw coordinates only works when the centre is (0, 0).

Method: 1. Use |k| for the size, and use the minus sign to put the image on the opposite side of the centre. 2. Scale factor -3 makes the image three times as far away. 3. Scale factor -1/2 makes the image half as far, still on the opposite side.

The vector from the centre

Use three lines for every point. First subtract the centre from the object point to get the vector. For a point (5, 1) and centre (2, 1) that vector is (3, 0). Second, multiply both components by the scale factor. With scale factor -1 the vector becomes (-3, 0). Third, add the centre back: (2, 1) + (-3, 0) = (-1, 1). The image is on the same horizontal line, the same distance from the centre, on the other side.

A second example with the centre at the origin is quicker. The point (2, -4) enlarged by scale factor -3 from (0, 0) goes to (-6, 12). Each coordinate is multiplied by -3. Check the distance: the object is √(2^2 + (-4)^2) from the origin and the image should be three times as far. If your image is closer, or still on the same side, the scale factor was not applied to the vector from the centre.

Method: 1. Subtract the centre, multiply by k, then add the centre back. 2. Point (5,1) from centre (2,1) with scale factor -1 goes to (-1,1). 3. From the origin, multiply the coordinates: (2,-4) with scale factor -3 goes to (-6,12).

Enlarging a whole shape

A shape is enlarged one vertex at a time, all from the same centre and all with the same scale factor. Join the image points in the same order as the object. For a triangle, find the three image vertices and draw the new triangle. The sides of the image are |k| times as long as the matching sides of the object. Angles stay the same, because an enlargement does not change angles. The image is similar to the object.

Place a ruler in your mind from an object vertex through the centre. The image vertex lies on that same straight line, on the other side of the centre when k is negative. That collinear check catches a point that has been plotted on the wrong side. The whole image is rotated 180^° relative to a positive enlargement. It is not a mirror image. If you have reflected the shape in a vertical or horizontal line, you have used a different transformation.

Method: 1. Repeat the same centre and the same k for every vertex. 2. Multiply each object length by |k|. 3. Check each image point lies on the ray through the centre, on the far side when k is negative.

Lengths and a final check

Lengths on the image equal object lengths multiplied by |k|. Scale factor -2 doubles every length. Scale factor -1 leaves every length unchanged, which is why it matches a half turn. Do not write a negative length. The minus sign has already been used to choose the side of the centre. Area uses k^2, and a square is positive, so the area scale factor is the same for k and for -k. This lesson only needs that fact as a check: a negative scale factor does not produce a negative area.

Finish by reading one point back. Distance from the centre to the image should equal |k| times the distance from the centre to the object. The centre should sit between the object point and the image point when k is negative, and the three points should be on one straight line. For scale factor -1, the centre is the midpoint of the object point and the image point. If those checks fail, recompute the vector before you join the shape.

Method: 1. Multiply lengths by |k|, never by the negative sign itself. 2. Scale factor -2 doubles lengths, and scale factor -1 leaves lengths unchanged. 3. For scale factor -1, the centre is the midpoint of a point and its image.

Describing a negative enlargement fully

A "describe fully" question shows two shapes and asks for the single transformation that maps one to the other. For an enlargement you need three things: the word enlargement, the scale factor and the centre. If the image is on the opposite side of the centre and upside down, the scale factor is negative.

To find the scale factor, compare a pair of matching lengths and give it a negative sign. To find the centre, draw a straight line through each point and its image. The lines meet at the centre of enlargement. You can also check numerically: the centre, a point and its image lie on one straight line, with the centre between them.

Method: 1. Check that the shapes are similar, one an upside-down copy of the other. 2. Divide a length on the image by the matching length on the object and make the answer negative. 3. Join two points to their images with straight lines and find where the lines cross. 4. Write: enlargement, scale factor and centre, with the centre as coordinates.

Tip: For a negative scale factor the centre lies between each point and its image.

Negative fractional scale factors

A scale factor such as -1/2 shrinks the shape to half size and also puts it on the opposite side of the centre. The negative sign gives the direction, and the size of the number gives the change in length. Image distances from the centre are half the object distances.

The vector method still works. From the centre to the point is a vector. Multiply that vector by -1/2, then add the result to the centre. For the point (6, 2) with centre (2, 2), the vector is beginpmatrix 4 \ 0 endpmatrix, times -1/2 gives beginpmatrix -2 \ 0 endpmatrix, so the image is (0, 2). Work out all the vertices this way and then join them to draw the image.

Method: 1. Subtract the centre from the point to get the vector from the centre. 2. Multiply the vector by the scale factor, including the negative sign. 3. Add the new vector to the centre to find the image point. 4. Repeat for each vertex and check that the shape is smaller and on the other side.

Lengths, areas and a negative scale factor

A negative scale factor still obeys the length rule: every length on the image is |k| times the matching length on the object. A side of 5 cm becomes 15 cm under a scale factor of -3, never -15 cm. Lengths and areas can never be negative. This is a common place to lose a mark if the sign is carried through by mistake.

Areas change by the square of the scale factor, so with k = -3 the area is multiplied by (-3)^2 = 9. Perimeters, being lengths, are multiplied by 3. Questions of this type are usually worth three or four marks: one for the size of the length or area factor, one for applying it, and one for the answer with the correct unit.

Method: 1. Ignore the sign for lengths: multiply by |k|. 2. For areas, multiply by k^2, which is always positive. 3. Apply the factor to the object value and state the units. 4. Check that the answer is larger than the object when |k| > 1 and smaller when |k| < 1.

Combining an enlargement with another transformation

A harder question enlarges a shape and then translates, reflects or rotates the result. Work step by step: write down the coordinates after the first transformation, then apply the second to those new coordinates. Label each shape so the steps are easy to follow. Keeping each stage in a small table of coordinates helps you avoid mixing up the shapes.

A scale factor of -1 gives the same image as a rotation of 180^° about the same centre. And an enlargement with scale factor -2 followed by a translation is another enlargement with scale factor -2 about a different centre. You can find the new centre from a point and its final image, using the same centre method as before.

Method: 1. Write the coordinates of the object. 2. Apply the enlargement using the vector method and record the image. 3. Apply the second transformation to the image. 4. For a single equivalent transformation, find the scale factor and centre from a point and its final image.

Remember

Multiply both coordinates by the scale factor.
A negative factor sends the point through the origin.
Image distance is the object distance times the size of the factor.

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