Describing Function Transformations

A complete exam description names the transformation and gives its vector, mirror line or scale factor and direction.

Write a complete description

Translation: give the vector $\binom{p}{q}$ .
Reflection: give the mirror line, such as the $x$ -axis.
Stretch: give the scale factor and say parallel to which axis.

For a quadratic, complete the square first. Then read the bracket as horizontal and the outside as vertical.

Worked example

Real exam: describe how $y=x^2$ maps onto $\mathrm{g}(x)=2x^2-8x+14$ (9709/12 February/March 2022 Q5b)

Complete the square so the bracket is clean: $2x^2-8x+14=2\big[(x-2)^2+3\big]=2(x-2)^2+6$ .
One two-step route is to translate first by $\tbinom{2}{3}$ , giving $y=(x-2)^2+3$ , and then stretch by scale factor $2$ parallel to the $y$ -axis. The later stretch doubles every output, including the earlier up-3 move.
An equally valid route is to stretch first by scale factor $2$ parallel to the $y$ -axis, giving $y=2x^2$ , and then translate by $\tbinom{2}{6}$ .

Answer

Route A: translate by

\tbinom{2}{3}

, then stretch vertically by factor 2.Route B: stretch vertically by factor 2, then translate by

\tbinom{2}{6}

Swipe left or right to see the whole diagram →

Both routes reach the same graph, but the vertical part of the translation changes because a later vertical stretch also stretches that earlier move.

(9709/12 February/March 2022 Q5(b)) These are the two routes listed in the published mark scheme. The translation vector changes with the order; mixing $\tbinom{2}{3}$ from one route with the order from the other gives the wrong graph.

A horizontal move and a vertical move may be stated in either order. Two moves in the same direction may need different numbers when the order changes; rebuild the equation to check.

Your turnIndependent exam-language check

Describe the translation that maps $y=x^2$ onto $y=x^2+6x+11$ .

Show worked answer

Complete the square: $x^2+6x+11=(x+3)^2+2$ . The $x+3$ moves the graph left 3 and the outside $+2$ moves it up 2.

Answer

\text{translation by }\binom{-3}{2}

PreviousThe Four Function Transformations NextCombined Function Transformations

Describing Function Transformations

A complete exam description names the transformation and gives its vector, mirror line or scale factor and direction.

Write a complete description

Translation: give the vector $\binom{p}{q}$ .
Reflection: give the mirror line, such as the $x$ -axis.
Stretch: give the scale factor and say parallel to which axis.

For a quadratic, complete the square first. Then read the bracket as horizontal and the outside as vertical.

Worked example

Real exam: describe how $y=x^2$ maps onto $\mathrm{g}(x)=2x^2-8x+14$ (9709/12 February/March 2022 Q5b)

Complete the square so the bracket is clean: $2x^2-8x+14=2\big[(x-2)^2+3\big]=2(x-2)^2+6$ .
One two-step route is to translate first by $\tbinom{2}{3}$ , giving $y=(x-2)^2+3$ , and then stretch by scale factor $2$ parallel to the $y$ -axis. The later stretch doubles every output, including the earlier up-3 move.
An equally valid route is to stretch first by scale factor $2$ parallel to the $y$ -axis, giving $y=2x^2$ , and then translate by $\tbinom{2}{6}$ .

Answer

Route A: translate by

\tbinom{2}{3}

, then stretch vertically by factor 2.Route B: stretch vertically by factor 2, then translate by

\tbinom{2}{6}

Swipe left or right to see the whole diagram →

Both routes reach the same graph, but the vertical part of the translation changes because a later vertical stretch also stretches that earlier move.

A horizontal move and a vertical move may be stated in either order. Two moves in the same direction may need different numbers when the order changes; rebuild the equation to check.

Your turnIndependent exam-language check

Describe the translation that maps $y=x^2$ onto $y=x^2+6x+11$ .

Show worked answer

Complete the square: $x^2+6x+11=(x+3)^2+2$ . The $x+3$ moves the graph left 3 and the outside $+2$ moves it up 2.

Answer

\text{translation by }\binom{-3}{2}

PreviousThe Four Function Transformations NextCombined Function Transformations