Transformations of graphs

What you are looking at

Each parameter does one job. $a$ stretches the graph vertically by scale factor $|a|$, $b$ stretches it horizontally by scale factor $\dfrac{1}{|b|}$, and $c$ and $d$ translate it by the vector $\begin{pmatrix} c \\ d \end{pmatrix}$. A negative $a$ reflects the graph in the $x$-axis and a negative $b$ reflects it in the $y$-axis. The visualiser draws the original as a dashed curve and the transformed graph in blue, and lists the transformations in words as you move the sliders.

Two key points, $A$ and $B$, are marked on the original graph, for example $(0, 0)$ and $(1, 1)$ on $y = x^2$. Dashed yellow lines join each one to its image $A'$ or $B'$, and the working panel applies the rule $(x, y) \mapsto \left(\dfrac{x}{b} + c,\ ay + d\right)$. For $y = e^x$ it also tracks the horizontal asymptote, which moves from $y = 0$ to $y = d$. Animate replays the change in stages: horizontal stretch, then vertical stretch, then translation.

Try this

1. 1With $f(x) = x^2$, set $c = 3$. The equation reads $y = (x-3)^2$ and the vertex moves to $(3, 0)$: a minus sign inside the bracket moves the graph to the right.
2. 2Choose $f(x) = \sin x$ and set $b = 2$. The graph is squeezed towards the $y$-axis, $B\left(\dfrac{\pi}{2}, 1\right)$ moves to $\left(\dfrac{\pi}{4}, 1\right)$, and the working names a horizontal stretch with scale factor $\dfrac{1}{2}$.
3. 3Choose $f(x) = e^x$, then set $a = -1$ and $d = 2$. The graph is reflected in the $x$-axis and the yellow asymptote moves up to $y = 2$.
4. 4Set all four sliders away from their defaults and press Animate. The graph passes through each stage in turn, showing the order in which the transformations act for this form of the equation.

Mistakes students make

• Moving $y = f(x-3)$ three units to the left. It moves 3 units to the right, because $x$ has to be 3 larger to give $f$ the same input as before.
• Reading $y = f(2x)$ as a stretch with scale factor 2. It is a horizontal stretch with scale factor $\dfrac{1}{2}$: every $x$-coordinate is halved.
• Translating $y = f(2x+4)$ by 4 units. Factorise first: $f(2x+4) = f\big(2(x+2)\big)$ is a horizontal stretch with scale factor $\dfrac{1}{2}$ followed by a translation 2 units to the left.

Where it is taught

• IB Maths AA SL 2.11 · HL 2.11 Transformations of graphs
• Edexcel A-Level Maths Pure Year 1 · Graphs and transformations
• Greek Lykeio (Β΄ Λυκείου) · Άλγεβρα, chapter 2: vertical and horizontal shifts of a graph