Translation, Scaling, and Reflection

The Big Idea

A transformation is a rule that maps each permitted input to an output. On the number line, three foundational transformations are translation, scaling, and reflection.

These rules are not isolated tricks. They are functions on \(\mathbb R\), and they become the building blocks for affine functions, graph transformations, coordinate geometry, unit conversion, computer graphics, calibration, and data analysis.

Three Transformations on the Number Line

The arrows identify input-output correspondence. A transformation applies the same rule to every point in its domain, not only to the example point displayed.

Inputs, Images, and Preimages

If a transformation \(F\) maps \(x\) to \(y\), we write

\[ F(x)=y. \]

The output \(y\) is the image of \(x\). The input \(x\) is a preimage of \(y\).

For \(F(x)=2x+1\),

\[ F(3)=7. \]

Thus \(7\) is the image of \(3\), and \(3\) is the preimage of \(7\). This vocabulary lets us discuss transformations of individual points, intervals, sets, graphs, and geometric objects with one language.

Translation Adds the Same Amount Everywhere

Translation by \(h\) is the rule

\[ T_h(x)=x+h. \]

• If \(h>0\), every point shifts right by \(h\) units.
• If \(h<0\), every point shifts left by \(|h|\) units.
• If \(h=0\), every point remains fixed.

Under \(T_4\), the points \(-3\), \(0\), and \(5\) map to \(1\), \(4\), and \(9\). Their coordinates change, but the distances between them do not.

Translation Preserves Distance and Order

For any real points \(a\) and \(b\),

\[ |T_h(b)-T_h(a)| =|(b+h)-(a+h)| =|b-a|. \]

Translation is therefore a distance-preserving transformation, also called an isometry.

It preserves order as well:

\[ a<b \quad\Longrightarrow\quad a+h<b+h. \]

Every point moves together, so the shape and orientation of any interval remain unchanged.

Scaling Multiplies Distance from the Origin

Scaling by \(k\) about the origin is

\[ S_k(x)=kx. \]

The output’s distance from zero is

\[ |S_k(x)|=|kx|=|k||x|. \]

A factor with \(|k|>1\) enlarges distance; \(0<|k|<1\) contracts distance; \(|k|=1\) preserves distance; and \(k=0\) collapses every point to the origin.

The Sign of a Scale Factor Controls Orientation

For \(a<b\), positive scaling gives \(ka<kb\), while negative scaling gives \(ka>kb\). The inequality reversal is the algebraic trace of reflecting left and right.

Reflection Across the Origin

Reflection across zero maps every point to its additive inverse:

\[ R(x)=-x. \]

The points \(x\) and \(-x\) have equal distance from the origin and lie on opposite sides:

\[ |x|=|-x|. \]

Reflection preserves distance but reverses order. It is the special scaling \(S_{-1}\).

Reflection Across Any Center

Reflection across a center \(c\) is

\[ R_c(x)=2c-x. \]

To understand the formula, subtract \(c\) to measure the point relative to the center, negate that displacement, and add \(c\) back:

\[ R_c(x)=c-(x-c)=2c-x. \]

The center is the midpoint between the input and its image:

\[ \frac{x+R_c(x)}2=c. \]

Reflecting \(8\) across \(c=3\) gives \(R_3(8)=6-8=-2\).

Scaling About a Center Other Than Zero

To scale about a chosen center \(c\), first measure displacement from \(c\), scale that displacement, and then restore the center:

\[ D_{c,k}(x)=c+k(x-c). \]

This rule keeps \(c\) fixed. For \(c=4\), \(k=2\), and \(x=7\),

\[ D_{4,2}(7)=4+2(7-4)=10. \]

The input was \(3\) units right of the center; the image is \(6\) units right. When \(k=-1\), centered scaling becomes reflection across \(c\).

What Each Transformation Preserves

Transforming a Set Means Transforming Every Member

If \(F\) is a transformation and \(A\subseteq\mathbb R\), the image of the set is

\[ F(A)=\{F(x):x\in A\}. \]

For \(A=[-2,3]\) and \(T_4(x)=x+4\),

\[ T_4(A)=[2,7]. \]

The rule applies to every real member of the interval, not only its two endpoints. The endpoints are enough to describe the image here because translation is continuous and order-preserving.

Interval Transformation Rules

Let \(a\le b\).

Open and mixed endpoint status travels with the endpoint’s image. After an order-reversing transformation, the left and right positions switch, so rewrite the interval in increasing order.

Bounds and Extrema Transform Too

If a nonempty bounded set \(A\) has infimum \(m\) and supremum \(M\), then translation gives

\[ \inf T_h(A)=m+h, \qquad \sup T_h(A)=M+h. \]

Positive scaling gives

\[ \inf S_k(A)=km, \qquad \sup S_k(A)=kM, \qquad k>0. \]

Negative scaling exchanges the lower and upper roles:

\[ \inf S_k(A)=kM, \qquad \sup S_k(A)=km, \qquad k<0. \]

Whether a sharp bound is attained also transfers under every reversible transformation.

Composition Builds Transformation Pipelines

To apply \(G\) first and then \(F\), use the composition

\[ (F\circ G)(x)=F(G(x)). \]

Suppose \(S_2(x)=2x\) and \(T_3(x)=x+3\). Scale first and then translate:

\[ (T_3\circ S_2)(x)=2x+3. \]

Translate first and then scale:

\[ (S_2\circ T_3)(x)=2(x+3)=2x+6. \]

The order matters. A transformation pipeline should be read from the inside outward.

Affine Transformations Combine Scaling and Translation

A number-line affine transformation has the form

\[ F(x)=ax+b. \]

In this written order, the rule first scales \(x\) by \(a\) and then translates by \(b\).

• \(|a|\) controls the distance scale.
• The sign of \(a\) controls orientation.
• \(b\) shifts the scaled line.
• If \(a=0\), the rule collapses every input to \(b\).

For \(F(x)=-2x+5\), the input \(3\) first scales and reflects to \(-6\), then translates right \(5\) to \(-1\).

What \(F(x)=ax+b\) Does to Distance and Order

For any inputs \(x_1\) and \(x_2\),

\[ |F(x_2)-F(x_1)| =|a||x_2-x_1|. \]

The translation term \(b\) cancels from the difference. Therefore distance scaling depends only on \(|a|\).

• If \(a>0\), \(F\) preserves order.
• If \(a<0\), \(F\) reverses order.
• If \(a=0\), \(F\) collapses all order information to one output.

Inverse Transformations Undo the Rule

For \(F(x)=ax+b\) with \(a\ne0\), solve \(y=ax+b\) for \(x\):

\[ F^{-1}(y)=\frac{y-b}{a}. \]

The inverse reverses the original steps in reverse order: subtract \(b\), then divide by \(a\).

A Reflection Undoes Itself

Apply the same reflection twice:

\[ R_c(R_c(x)) =2c-(2c-x) =x. \]

A transformation that is its own inverse is called an involution. Negation, reciprocal on a nonzero domain, and many geometric reflections provide important examples.

Fixed Points Stay Where They Are

A fixed point satisfies

\[ F(x)=x. \]

For \(F(x)=ax+b\) with \(a\ne1\), the fixed point is

\[ x=\frac{b}{1-a}. \]

Reversibility Requires Distinct Inputs to Stay Distinct

Translation, reflection, and nonzero scaling map distinct real inputs to distinct outputs. Each output has exactly one preimage, so the transformation can be reversed.

The constant rule \(F(x)=b\) sends every input to the same output. It loses the input information and has no inverse function on \(\mathbb R\).

Thus the affine rule \(F(x)=ax+b\) is a one-to-one, onto transformation of \(\mathbb R\) exactly when \(a\ne0\).

Moving Points and Changing Coordinates Are Different Views

An active transformation moves or replaces points while the coordinate system remains fixed. A translation \(x\mapsto x+3\) sends the point at \(2\) to the point at \(5\).

A passive coordinate change keeps the physical point fixed but changes the reference system or labels. Moving the origin right by \(3\) would make the old coordinate \(5\) receive new coordinate \(2\).

The two descriptions are closely related but use opposite coordinate adjustments. State whether objects are moving or the measuring system is changing.

Temperature Conversion Is Affine

Celsius-to-Fahrenheit conversion uses

\[ F=\frac95C+32. \]

The factor \(9/5\) changes the unit interval, while \(32\) shifts the zero reference. This is not a pure scaling because \(0^\circ\text{C}\ne0^\circ\text{F}\).

The inverse transformation is

\[ C=\frac59(F-32). \]

The inverse subtracts the offset first and then reverses the scale.

Calibration Corrects Scale and Offset

A sensor may report a raw value \(x\) that is consistently stretched and shifted relative to the true measurement. An affine calibration

\[ y=ax+b \]

corrects the scale with \(a\) and the offset with \(b\). Two accurate reference pairs with distinct raw inputs determine a nonconstant affine calibration.

A negative \(a\) indicates that larger raw readings correspond to smaller calibrated values, reversing order.

Centering and Normalizing Data

Subtracting a reference value centers data. Dividing by a positive scale standardizes the unit:

\[ z=\frac{x-\mu}{\sigma}, \qquad \sigma>0. \]

Here \(\mu\) is the chosen center and \(\sigma\) is the scale. The transformation translates by \(-\mu\) and then scales by \(1/\sigma\).

The inverse returns to the original unit:

\[ x=\mu+\sigma z. \]

Maps and Computer Graphics Use Transformation Pipelines

A map or display may translate coordinates to a new origin, scale them to fit a viewport, and reflect one axis because screen coordinates often increase downward.

The order of these steps changes the result. Graphics systems therefore store transformations in an explicit pipeline and often represent affine transformations with matrices in higher dimensions.

The one-dimensional rule \(ax+b\) is the simplest working model for that larger machinery.

A Preview of Function-Graph Transformations

Number-line transformations can act on input coordinates, output coordinates, or both. If \(y=f(x)\), then

• \(y=f(x-h)\) shifts the graph horizontally by \(h\);
• \(y=f(x)+k\) shifts it vertically by \(k\);
• \(y=af(x)\) scales output values by \(a\);
• \(y=f(bx)\) changes input spacing by reciprocal factor \(1/|b|\).

Inside and outside transformations act on different coordinate lines, which is why their visual effects differ. Later graphing lessons can develop these rules in detail.

Check a Transformation with Structure

A single substituted point can verify arithmetic, but structural checks test the whole rule.

• A translation should preserve every distance and order relation.
• A scaling by \(k\) should multiply every distance by \(|k|\).
• A negative scale or reflection should reverse order.
• A reflection across \(c\) should place \(c\) at the midpoint of each input-image pair.
• A proposed inverse should return every tested input: \(F^{-1}(F(x))=x\).

For an affine rule, test at least two distinct points. One point can confirm an offset but cannot reveal whether the scale is correct.

A Reliable Transformation Routine

1. Identify the input and domain.
2. Write the rule as a function. Keep translation, scaling, and reflection roles visible.
3. Follow composition order. Work from the inside outward.
4. Track the reference point. Scaling and reflection require a center.
5. Check orientation. A negative scale reverses order.
6. Transform sets completely. Map every member, not only a convenient sample.
7. Restore interval order. Place the lesser image boundary on the left.
8. Find the inverse when needed. Reverse the steps in reverse order.
9. Verify a preserved property. Check distance, midpoint, order, or fixed points.

Common Confusions

Working Summary

Translation adds a constant, scaling multiplies distance from a center, and reflection reverses position across a center. Each is a function that maps inputs, points, intervals, and sets to images.

Translation preserves distance and order. Scaling multiplies distance by \(|k|\), preserves order for \(k>0\), and reverses it for \(k<0\). Reflection preserves distance, reverses order, and undoes itself.

Composition builds affine rules \(ax+b\), and order matters. Nonzero affine transformations are reversible, transform bounds and intervals predictably, and support unit conversion, calibration, normalization, graphics, graph transformations, and higher-dimensional geometry.

Check Your Understanding