Starting from \(y=f(x)\)
A transformation takes a graph you already know, the "parent" curve \(y = f(x)\), and moves or reshapes it. The clever part is that a small change to the equation produces a predictable change to the picture. Master the pattern and you can sketch a family of curves from one known shape.
There is one habit worth building early: changes on the outside of \(f\) affect the graph vertically (the \(y\) direction), and they behave as you expect. Changes on the inside, next to the \(x\), affect the graph horizontally, and they behave in the opposite way to what you first guess.
Where this is assessed
Naming a transformation correctly and describing it fully is Criterion A and Criterion C work. Applying a transformation to model a shifted or scaled situation, and interpreting it, reaches Criterion D.
Translations: \(f(x)+a\) and \(f(x+a)\)
A translation slides the whole graph without turning or resizing it. There are two kinds, and the position of the number in the equation tells you which.
- \(f(x) + a\): adding \(a\) on the outside slides the graph up by \(a\) (or down if \(a\) is negative). This is the intuitive one.
- \(f(x + a)\): adding \(a\) on the inside slides the graph left by \(a\). This feels backwards: \(+a\) inside moves it in the negative \(x\) direction, and \(-a\) inside moves it right.
| Equation | Move | Vector |
|---|---|---|
| \(f(x) + a\) | Up by \(a\) | \(\begin{pmatrix} 0 \\ a \end{pmatrix}\) |
| \(f(x) - a\) | Down by \(a\) | \(\begin{pmatrix} 0 \\ -a \end{pmatrix}\) |
| \(f(x + a)\) | Left by \(a\) | \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\) |
| \(f(x - a)\) | Right by \(a\) | \(\begin{pmatrix} a \\ 0 \end{pmatrix}\) |
The point \((2,\ 5)\) lies on \(y = f(x)\). Where does it move to on \(y = f(x - 1) + 3\)?
Inside is backwards
The most common slip is moving \(f(x + 3)\) to the right. It goes left. If it helps, ask "what \(x\) makes the bracket zero?" For \(x + 3\) that is \(x = -3\), which is where the old \(x = 0\) feature now sits: to the left.
Reflections in the axes
A reflection flips the graph like a mirror image. Again the position of the minus sign tells you which axis is the mirror.
- \(-f(x)\): the minus is on the outside, so every \(y\)-value changes sign. The graph flips upside down, reflected in the \(x\)-axis.
- \(f(-x)\): the minus is on the inside, so every \(x\)-value changes sign. The graph flips left to right, reflected in the \(y\)-axis.
The pattern lines up with translations: outside acts on \(y\), inside acts on \(x\).
Stretches: vertical and horizontal
A stretch resizes the graph in one direction, pulling it away from or squashing it towards an axis.
- \(a\,f(x)\): multiplying the whole function by \(a\) is a vertical stretch of scale factor \(a\). Every \(y\)-value is multiplied by \(a\), so the curve is pulled away from the \(x\)-axis (or squashed towards it if \(0 < a < 1\)).
- \(f(ax)\): multiplying the input by \(a\) is a horizontal stretch of scale factor \(\dfrac{1}{a}\). The curve is squashed towards the \(y\)-axis when \(a > 1\). Once more, the inside change behaves as the reciprocal of what it looks like.
Starting from \(f(x) = x^2\), describe fully the single transformation that gives \(y = 2f(x)\), and state the new equation.
Say the whole description
For full marks a transformation needs a type and its detail: "translation by the vector \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\)", or "vertical stretch, scale factor \(2\)", not just "it moves up" or "it gets bigger".
Check yourself
Describe each transformation fully, then check against the answer.
1. Describe the transformation from \(y = f(x)\) to \(y = f(x) - 4\). +
The \(-4\) is on the outside, so it changes the \(y\)-values. It is a translation down by 4, that is by the vector \(\begin{pmatrix} 0 \\ -4 \end{pmatrix}\).
2. The point \((4,\ -2)\) lies on \(y = f(x)\). Where is its image on \(y = f(-x)\)? +
\(f(-x)\) is a reflection in the \(y\)-axis, so the \(x\)-coordinate changes sign while \(y\) stays the same. The image is \((-4,\ -2)\).
3. Describe the transformation from \(y = f(x)\) to \(y = f(2x)\). +
The \(2\) multiplies the input, so it is a horizontal change with the reciprocal scale factor. It is a horizontal stretch of scale factor \(\tfrac{1}{2}\), squashing the graph towards the \(y\)-axis.
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