Graph transformationsEdexcel A-Level Maths: Revision notes

• $y=f(x)+a$ is a translation by $\binom{0}{a}$: every point moves up $a$ (down if $a<0$).
• $y=f(x+a)$ is a translation by $\binom{-a}{0}$: every point moves left $a$. Example: $y=x^2-6x+5=(x-3)^2-4$ is $y=x^2$ translated by $\binom{3}{-4}$. A point $(3,-2)$ on $y=f(x)$ moves to $(1,-2)$ on $y=f(x+2)$ and to $(3,0)$ on $y=f(x)+2$. Asymptotes move too: $y=\frac1x+3$ has asymptotes $x=0$ and $y=3$; $y=\frac{1}{x+2}$ has $x=-2$ and $y=0$.

• $y=af(x)$ is a stretch parallel to the $y$-axis with scale factor $a$: $y$-coordinates are multiplied by $a$ and $x$-intercepts stay fixed.
• $y=f(ax)$ is a stretch parallel to the $x$-axis with scale factor $\frac1a$: $x$-coordinates are divided by $a$ and $y$-intercepts stay fixed. Example: $(3,-2)$ on $y=f(x)$ goes to $(3,-4)$ on $y=2f(x)$ and to $\left(\frac32,-2\right)$ on $y=f(2x)$. For $y=\sin x$, $y=3\sin x$ has amplitude $3$ and $y=\sin2x$ has period $\pi$.

• $y=-f(x)$ is a reflection in the $x$-axis: $y$-coordinates change sign.
• $y=f(-x)$ is a reflection in the $y$-axis: $x$-coordinates change sign. Example: reflecting $y=x^2-6x+5$ in the $x$-axis gives $y=-x^2+6x-5$; in the $y$-axis gives $y=x^2+6x+5$. A reflection in the $x$-axis turns a maximum into a minimum, and $y=e^{-x}$ is the reflection of $y=e^x$ in the $y$-axis, with the same asymptote $y=0$.

For $y=af(bx+c)+d$ work in the order you would evaluate it: the brackets first (changes to $x$), then the multiplier $a$, then add $d$. Example: $y=2f(-x)+1$. Reflect in the $y$-axis, stretch by factor $2$ parallel to the $y$-axis, then translate up $1$. The point $(3,-2)$ becomes $(-3,-2)$, then $(-3,-4)$, then $(-3,-3)$. Example: $y=3+\cos2x$ is $y=\cos x$ stretched by factor $\frac12$ parallel to the $x$-axis, then translated by $\binom{0}{3}$. Maximum $4$, minimum $2$, period $\pi$, $y$-intercept $(0,4)$. Order matters when a stretch and translation act in the same direction: $2f(x)+1$ is not $2(f(x)+1)=2f(x)+2$.

• Quadratics, cubics, quartics: track turning points and intercepts. If $f(x)=x^3-4x$ meets the axis at $-2,0,2$, then $f(x-1)$ meets it at $-1,1,3$, and $-f(x/2)$ has roots $0,\pm4$.
• Reciprocals $\frac ax$, $\frac a{x^2}$: transform the asymptotes as well as the curve, e.g. $y=\frac1{x+2}+3$ has $x=-2$ and $y=3$.
• Trigonometric $\sin x$, $\cos x$, $\tan x$: stretches change amplitude and period; $y=-\cos\left(x+\frac\pi4\right)$ is $y=\cos x$ translated $\frac\pi4$ left and reflected in the $x$-axis. $\tan x$ has asymptotes at $x=\pm\frac\pi2$, which move under $f(ax)$ and $f(x+a)$.
• Exponentials $e^x$, $a^x$: $y=e^x+c$ has asymptote $y=c$; $y=e^{2x}$ keeps its $y$-intercept $(0,1)$ but grows faster; $y=e^{-x}$ decays. On a sketch, label intercepts, turning points and asymptotes with their new coordinates.