Graphing Functions using Transformations

"A picture is worth a thousand words" -- rather than plotting many points from scratch, it is far faster to recognise a complicated curve as a transformation of a simpler, already-known one. This chapter uses three tools: reflection, translation, dilation. Reflection and translation preserve size and shape (the new graph is congruent to the original); dilation changes the shape by stretching or compressing it.

Reflection. The reflection of $y=f(x)$ in a line $\ell$ is its mirror image across $\ell$. Three reflections recur constantly:

• $y=-f(x)$ is the reflection of $y=f(x)$ about the $x$-axis (e.g. $y=-x^2$ mirrors $y=x^2$ downward).
• $y=f(-x)$ is the reflection about the $y$-axis (e.g. $y=\sqrt{-x}$, defined for $x\le0$, mirrors $y=\sqrt x$ leftward).
• $y=f^{-1}(x)$ is the reflection about the line $y=x$ (e.g. $y=e^x$ and $y=\ln x$ are reflections of each other in $y=x$, since each is the other's inverse).

Translation. A translation shifts the graph without changing its shape:

E.g. from $f(x)=|x|$: $f(x-1)=|x-1|$ shifts the V-graph right by 1; $f(x+1)=|x+1|$ shifts it left by 1; $f(x)-1$ shifts it down 1; $f(x)+1$ shifts it up 1.

Dilation. Multiplying $f$ by a positive constant stretches/compresses the graph vertically (moves it away from or toward the $x$-axis): a multiplier $>1$ stretches away from the $x$-axis; a multiplier $<1$ (but $>0$) compresses toward it. E.g. from $f(x)=x^2$: $\tfrac12x^2$ looks "flatter" (compressed toward the $x$-axis, multiplier $\tfrac12<1$), while $2x^2$ looks "narrower" (stretched away from the $x$-axis, multiplier $2>1$). A horizontal dilation instead rescales the $x$-axis: comparing $y=x^2-1$, $y=4(x^2-1)$ (vertical stretch by 4) and $y=(4x)^2-1$ (horizontal compression by factor 4 -- same $y$-intercept, but the curve looks "thinner" since $x$-values are scaled down by $\tfrac14$ to produce the same output).

The same idea extends to trigonometric curves: $y=\sin x$ and $y=\sin2x$ share the same maximum/minimum values ($\pm1$) but different $x$-intercepts ($\pm n\pi$ versus $\pm\tfrac12n\pi$) -- a horizontal compression by factor 2. …