Transformations of functionsIB MYP Maths Extended: Revision notes

Adding $a$ outside the function, $y=f(x)+a$, translates the graph up by $a$ (down if $a$ is negative). Every $y$-coordinate changes; the $x$-coordinates stay. Adding $a$ inside, $y=f(x+a)$, translates the graph left by $a$ (right if $a$ is negative). Every $x$-coordinate changes by $-a$. Example: if $f$ has a minimum at $(2,-1)$, then $f(x)+3$ has its minimum at $(2,2)$ and $f(x-4)$ has its minimum at $(6,-1)$.

$y=-f(x)$ reflects the graph in the $x$-axis: each $y$-coordinate changes sign, so $(4,6)$ becomes $(4,-6)$, and a minimum becomes a maximum. $y=f(-x)$ (the case $k=-1$ in $f(kx)$) reflects the graph in the $y$-axis: each $x$-coordinate changes sign.

$y=kf(x)$ is a vertical stretch, scale factor $k$, from the $x$-axis: every $y$-coordinate is multiplied by $k$. So $(4,6)$ on $y=f(x)$ becomes $(4,12)$ on $y=2f(x)$. $y=f(kx)$ is a horizontal stretch, scale factor $\frac1k$, from the $y$-axis: every $x$-coordinate is divided by $k$. So $(4,6)$ becomes $(2,6)$ on $y=f(2x)$, and $f\left(\frac{x}{2}\right)$ stretches horizontally by factor $2$. Points on the axis you stretch from do not move.

Apply the transformations to key points one at a time. For $y=-f(x+1)$: the translation first, $(4,6)\to(3,6)$, then the reflection, $(3,6)\to(3,-6)$. To describe a transformation, name its type (translation, reflection, stretch) and give the details: the vector, the axis of reflection, or the scale factor and direction. Compare $y=x^2$ and $y=(x+3)^2-5$: a translation with vector $\begin{pmatrix} -3 \\ -5 \end{pmatrix}$, so the vertex moves from $(0,0)$ to $(-3,-5)$.

To sketch a transformed graph, mark the key points of the original (intercepts, turning points, end points), move each according to the rule, and join them with the same shape. In context, a transformation changes the model: $y=f(x)+4$ lifts a bridge arch $4$ m, $1.5f(x)$ makes it $1.5$ times as tall, $f(\frac x2)$ makes it twice as wide. Check the new equation against known points such as the ground or the piers.