Transformations of graphsAQA A-Level Maths: Revision notes

A translation slides a graph without changing its shape.

• $y=f(x)+a$ moves the graph up by $a$ (down if $a<0$): vector $\begin{pmatrix}0\\ a\end{pmatrix}$.
• $y=f(x+a)$ moves the graph left by $a$ (right if $a<0$): vector $\begin{pmatrix}-a\\ 0\end{pmatrix}$. If $(3,-2)$ is a minimum of $y=f(x)$, then $y=f(x)+4$ has its minimum at $(3,2)$ and $y=f(x+4)$ at $(-1,-2)$. Changes outside the brackets act on $y$ in the obvious way; changes inside the brackets act on $x$ the opposite way.

A stretch changes size in one direction.

• $y=af(x)$: stretch parallel to the $y$-axis with scale factor $a$. Every $y$-coordinate is multiplied by $a$; $x$-intercepts do not move.
• $y=f(ax)$: stretch parallel to the $x$-axis with scale factor $\frac1a$. Every $x$-coordinate is divided by $a$; the $y$-intercept does not move. For example, $f(2x)$ halves the $x$-coordinates: roots $-2$ and $3$ become $-1$ and $\frac32$. $f\left(\frac x2\right)$ doubles them.

Choosing $a=-1$ in $af(x)$ gives $y=-f(x)$, a reflection in the $x$-axis: each $y$-coordinate changes sign, so maxima become minima. The $x$-intercepts stay put. If $y=f(x)$ has a maximum at $(2,6)$ and crosses the $y$-axis at $(0,4)$, then $y=-f(x)$ has a minimum at $(2,-6)$ and crosses the $y$-axis at $(0,-4)$. The sign of $a$ in $af(x)$ decides whether the graph is flipped; its size decides the stretch.

The safest method is to track key points. For each point $(x,y)$ on $y=f(x)$:

• $y=f(x)+a$: $(x,y+a)$
• $y=f(x+a)$: $(x-a,y)$
• $y=af(x)$: $(x,ay)$
• $y=f(ax)$: $\left(\frac xa,y\right)$ Example: if $f(x)=x^2-4x+7=(x-2)^2+3$ has minimum $(2,3)$ and $y$-intercept $(0,7)$, then $y=f(2x)$ has minimum $(1,3)$ and the same $y$-intercept $(0,7)$. Check: $f(2x)=4x^2-8x+7$, with minimum at $x=1$.

To describe a transformation fully, name the type and give its details: a translation needs a vector, a stretch needs a scale factor and a direction (parallel to which axis), a reflection needs a line. To sketch the transformed graph:

1. Mark the key points of the original (intercepts, turning points, asymptotes).
2. Move each key point using the rules above.
3. Redraw the same shape through the new points, labelling coordinates. Translations and stretches by themselves do not change the shape's turning-point nature (a minimum stays a minimum), unless $a<0$.