Reading a transformation

A graph transformation changes a graph by moving its points systematically or altering their coordinates. Start with

Two viewpoints are useful:

• A change outside $f$, such as adding to or multiplying $f(x)$, acts on the outputs, so it changes vertical coordinates.
• A change inside $f$, such as replacing $x$ by $x-a$ or $qx$, acts on the inputs and changes horizontal coordinates. These inside changes appear to work in the opposite direction from the sign or multiplier shown.

The same rules apply to polynomial, rational, exponential, logarithmic, trigonometric and other functions. They also work for unfamiliar functions given in a modelling context. If the full equation is awkward to handle, transform recognizable points, intercepts, turning points, endpoints and asymptotes instead.

The graph below compares the main transformations of one non-symmetric parent function. Using a non-symmetric function makes the reflections in the two axes clearly different.

Translations

A translation moves every point through the same directed distance. It doesn’t alter the graph’s shape or orientation. The two rules are

and

$$y=f(x-a)$$

In $y=f(x)+b$, every output increases by $b$. The graph moves up if $b>0$ and down if $b<0$. In coordinate form,

$$(x,y)\mapsto(x,y+b)$$

For $y=f(x-a)$, a positive $a$ moves the graph right; a negative $a$ moves it left. The reversed-looking sign deserves a moment’s thought. The new graph reaches the old output $f(x)$ when its displayed input lies $a$ units further to the right. Therefore,

$$(x,y)\mapsto(x+a,y)$$

Combining both changes gives

$$y=f(x-a)+b$$

which is a translation by

$$\begin{pmatrix} a\\ b \end{pmatrix}$$

The upper entry gives the horizontal movement, while the lower entry gives the vertical movement. For example, a positive upper entry and a negative lower entry mean right and down.

Reflections

A reflection replaces every point with its mirror image in a specified line.

Reflecting in the $x$-axis reverses the sign of each output:

$$y=-f(x)$$

so the point mapping is

$$(x,y)\mapsto(x,-y)$$

The $x$-axis is invariant: any point on it stays fixed during the transformation. The $y$-axis maps onto itself as a set.

For reflection in the $y$-axis, the sign of every input changes:

$$y=f(-x)$$

with point mapping

$$(x,y)\mapsto(-x,y)$$

Here, the $y$-axis is pointwise invariant and the $x$-axis maps onto itself. When the original graph already has the relevant symmetry, its reflection may coincide with it.

Vertical and horizontal stretches

A scale factor is a dimensionless multiplier. It gives the ratio of a transformed distance from an invariant axis to the original distance.

A vertical stretch multiplies every vertical coordinate and leaves the horizontal coordinates unchanged:

$$(x,y)\mapsto(x,py)$$

When $p>1$, vertical distances from the $x$-axis increase. When $0<p<1$, they decrease; this is often called a vertical compression. The $x$-axis remains invariant. A negative multiplier combines a vertical stretch of scale factor $|p|$ with reflection in the $x$-axis.

A horizontal stretch multiplies every horizontal coordinate without changing the vertical coordinates. Write it as

$$(x,y)\mapsto\left(\frac{x}{q},y\right)$$

The reciprocal is the key detail. If $q=4$, horizontal distances become one quarter as large; the graph is not stretched by a factor of $4$. For $0<q<1$, the graph becomes wider. A negative $q$ also produces a reflection in the $y$-axis. Under horizontal scaling, the $y$-axis is invariant.

For example, changing $y=\cos x$ into $y=-2\cos(4x)$ combines a horizontal stretch with scale factor $\frac14$, a vertical stretch with scale factor $2$, and reflection in the $x$-axis. These principles apply to every permitted function family, not only trigonometric graphs.

Composite transformations and their order

A composite transformation applies two or more individual transformations in sequence. Order matters whenever one operation changes the quantity used by a later operation.

Consider

$$y=2f(x-3)-1$$

Beginning with $y=f(x)$, translate the graph right by $3$, apply a vertical stretch with scale factor $2$, then translate it down by $1$. The stretch does not include the final vertical translation. If the downward translation came before the stretch, the output would instead contain

$$2\bigl(f(x-3)-1\bigr)=2f(x-3)-2$$

which produces a different graph.

A practical approach is to read the expression by its structure:

• Deal with changes to the input of $f$ when locating horizontal positions.
• Evaluate or preserve the transformed function value.
• Apply outside multiplication to the output.
• Apply outside addition last.

Take care with horizontal expressions too. In $f(q(x-a))$, the factor $q$ multiplies the entire displacement $x-a$. This does not generally give the same transformation as $f(qx-a)$. The brackets show exactly which operation happens first.

The figure compares two sequences that contain the same vertical stretch and translation, but in opposite orders. Since their final graphs differ, a list of transformations can be incomplete if no order is given.

Transformations in modelling

Within a model, a transformation affects the meaning of its parameters as well as the graph’s appearance. A horizontal translation may represent a delayed event or a changed reference time. A vertical translation may show a new baseline, while a stretch may represent a change in magnitude or time scale. Units must match the appropriate axis: horizontal displacement uses the input variable’s unit, and vertical displacement uses the output variable’s unit. Scale factors are dimensionless.

You can often transform useful features directly. Translations move intercepts, turning points and asymptotes by the translation vector. A vertical stretch multiplies output coordinates and horizontal asymptote levels by $p$. A horizontal stretch multiplies input coordinates and vertical asymptote locations by $\frac1q$. As a result, the domain and range need to be updated rather than copied automatically.

A graphing utility can check a transformation by displaying the original and transformed functions together. Choose a viewing window that contains the transformed features. Otherwise, a correct translation or stretch may make the graph seem to disappear.

Connections

Translating a curve so that calculations use values close to a convenient local origin can reduce rounding error when the original input values are very large. The mathematical relationship stays the same, but a calculator or computer no longer has to carry unnecessary leading digits through intermediate work.

In economics, translations can represent shifts of supply or demand curves when an external condition changes the amount supplied or demanded at every price. In physics, translated or reflected curves can model changes in reference level, direction or timing in electromagnetic induction. The variables on the axes determine the precise interpretation, not the visual movement by itself.

Transformation rules are internally consistent across cultures. However, people choose the model, coordinate system, scale and graphical convention. This distinction helps address the knowledge question of whether mathematics is independent of culture: formal relationships may transfer widely, while cultural priorities and assumptions can shape the questions selected, the representations used and the meanings given to parameters.