18.3 Enlargement with fractional and negative scale factors — Learn

Prerequisites: enlargement, fractions, negative numbers and coordinates.

Objectives: use fractional and negative scale factors for enlargements about the origin.

A fractional scale factor between $0$ and $1$ makes a smaller image. A negative scale factor places the image on the opposite side of the centre.

For scale factor $k$ about the origin, map $(x,y)$ to $(kx,ky)$. The sign gives the side of the centre; the magnitude gives the length multiplier.

Example. Enlarge $(6,-4)$ by scale factor $\frac12$.

Multiply both coordinates by $\frac12$: $\left(3,-2\right)$. The image is $(3,-2)$ and half as far from the origin.

Example. Enlarge $(2,3)$ by scale factor $-2$.

Multiply $2$ by $-2$: $-4$. Multiply $3$ by $-2$: $-6$. The image is $(-4,-6)$, twice as far from the origin on the opposite side.

Watch out. A negative scale factor does not make the image lengths negative. Lengths use the positive magnitude $|k|$; the negative sign changes direction through the centre.

Progress check

A tile corner is at $(4,2)$. Work out its image under scale factor $-\frac12$ about the origin. Check both coordinates, then explain why the image lies on the opposite side of the origin.

You should now be able to

• enlarge points with fractional scale factors;
• enlarge points with negative scale factors;
• explain how sign and magnitude affect an image;
• find a scale factor from corresponding coordinates.