2.8 Transformations of graphsIB Maths: Applications and Interpretation HL: Revision notes

A translation slides the whole graph without turning or resizing it.

• $y=f(x)+b$ moves the graph $b$ units up (down if $b<0$).
• $y=f(x-a)$ moves the graph $a$ units to the right (left if $a<0$). The translation by the vector $\begin{pmatrix}3\\ -2\end{pmatrix}$ is 3 units right and 2 units down, giving $y=f(x-3)-2$. Each point $(x,y)$ moves to $(x+3,y-2)$. Example: $f(x)=x^2$ translated by $\begin{pmatrix}3\\ -2\end{pmatrix}$ gives $y=(x-3)^2-2$, with vertex $(3,-2)$.

• $y=-f(x)$ is a reflection in the $x$-axis: every $y$-coordinate changes sign, so $(x,y)\to(x,-y)$.
• $y=f(-x)$ is a reflection in the $y$-axis: every $x$-coordinate changes sign, so $(x,y)\to(-x,y)$. Example: if $P(4,6)$ lies on $y=f(x)$, then $(4,-6)$ lies on $y=-f(x)$ and $(-4,6)$ lies on $y=f(-x)$. Points on the line of reflection do not move.

• $y=p\,f(x)$ is a vertical stretch with scale factor $p$ (parallel to the $y$-axis): each $y$-coordinate is multiplied by $p$.
• $y=f(qx)$ is a horizontal stretch with scale factor $\frac1q$ (parallel to the $x$-axis): each $x$-coordinate is divided by $q$. Points on the $x$-axis are invariant (do not move) under a vertical stretch, and points on the $y$-axis are invariant under a horizontal stretch. Example: $y=\sin x$ to $y=4\sin 2x$ is a vertical stretch with scale factor 4 and a horizontal stretch with scale factor $\frac12$. The amplitude becomes 4 and the period becomes $\frac{2\pi}{2}=\pi$. Under $y=f(3x)$ the point $(4,6)$ moves to $\left(\frac43,6\right)$.

A composite transformation is two or more transformations applied one after another. The order can change the result. Example: $y=x^2$ to $y=3x^2+2$ is a vertical stretch with scale factor 3 followed by a translation by $\begin{pmatrix}0\\ 2\end{pmatrix}$. Doing the translation first gives $3(x^2+2)=3x^2+6$, a different graph. Another example: $y=\sin x$ to $y=4\sin 2x+1$ uses two stretches (any order) and then a translation up 1. To find the equation, build it up in the order described: after each step, write the new equation and use it as the starting point for the next. Two transformations in different directions (for example a horizontal translation and a vertical stretch) can be applied in either order; two in the same direction cannot.

Transformations let you adapt a known model. In $g(t)=f(t-2)$, the graph of $f$ is moved 2 units to the right, so every value of $f$ happens 2 hours (or weeks) later. In $y=f(x)+b$, every value rises by $b$. A vertical stretch $y=pf(x)$ scales every output, for example doubling the population at every time. Worked example: if $f(t)=2^t$ then $3f(t-2)=3\times2^{t-2}=\frac34\times2^t$, so shifting right by 2 and stretching by 3 is the same as a single vertical stretch with scale factor $\frac34$. Check by substituting a point: if $(1,5)$ is on $y=f(x)$ and the translation is $\begin{pmatrix}3\\ -2\end{pmatrix}$, then $(4,3)$ should be on the new graph.