FUNCTIONS 13
Transformations of Functions
Transformations move, stretch, compress or reflect a graph while preserving its underlying structure.
The main idea
Starting with \(y=f(x)\), a transformed function can often be written as
\[\boxed{y=A f(B(x-C))+D}.\]The constants outside \(f\) act vertically; the constants inside \(f\) act horizontally.
1. Vertical translation
\(y=f(x)+a\) moves the graph vertically. Compare \(y=x^2\) and \(y=x^2+3\).
2. Horizontal translation
\(y=f(x-a)\) shifts the graph right by \(a\). Thus \(y=(x-2)^2\) has vertex \((2,0)\).
3. Vertical stretching and compression
\(y=Af(x)\) multiplies every vertical distance by \(|A|\).
4. Reflection in the \(x\)-axis
\(y=-f(x)\) changes each point \((x,y)\) into \((x,-y)\).
5. Reflection in the \(y\)-axis
\(y=f(-x)\) changes each point \((x,y)\) into \((-x,y)\).
6. Horizontal stretching and compression
For \(y=f(Bx)\), the horizontal scale factor is \(1/|B|\). Compare \(\sin x\) and \(\sin(2x)\).
7. Several transformations together
For \(y=2(x-3)^2+1\), shift right \(3\), stretch vertically by \(2\), then shift up \(1\). The vertex is \((3,1)\).
8. Why inside transformations look reversed
For \(y=f(x-2)\), the original input \(0\) occurs when \(x-2=0\), so the feature moves to \(x=2\).
↓
\(x=2\)
shift right by 2
9. Reading the general transformation
In \(y=A f(B(x-C))+D\), \(A\) controls vertical scale/reflection, \(B\) horizontal scale/reflection, \(C\) horizontal shift, and \(D\) vertical shift.
\(B\): horizontal scale
\(C\): horizontal shift
\(D\): vertical shift
10. Why transformations matter for calculus
Transformations help us predict how parameters alter roots, turning points and rates of change.
↓ transformation
new graph
Outside changes act vertically and inside changes act horizontally. Translations move graphs, scale factors stretch or compress them, and negative factors produce reflections.