Translations, Reflections, Stretches, and Compressions

The Central Idea

A transformation changes the position, orientation, or scale of a graph while preserving its basic family structure.

If the parent function is

a large class of transformed functions can be written as

The parameters \(a\), \(b\), \(h\), and \(k\) control the transformation.

Four Basic Transformation Types

• translation — moves a graph without changing its shape;
• reflection — flips a graph across a line;
• stretch — enlarges distances in one direction;
• compression — reduces distances in one direction.

Transformations Preserve Family Structure

A transformed quadratic function is still quadratic. A transformed absolute-value function is still in the absolute-value family. A transformed square-root function retains the characteristic one-sided shape of the square-root family.

Transformations change the graph's placement or scale, not its underlying function family.

Start with a Parent Graph

A parent function is a simple reference model for a function family. Examples include

Instead of constructing every related graph independently, we can transform known points from the parent graph.

A Point-by-Point View

Suppose the point

lies on the graph of \(y=f(x)\). This means

A transformation tells us where that point moves on the new graph.

The General Coordinate Mapping

For

a point \((u,v)\) on \(y=f(x)\) moves to

This one rule contains translations, reflections, stretches, and compressions.

Why the Coordinate Rule Works

To make the transformed function use the same parent input \(u\), require

Solving for \(x\) gives

Since \(f(u)=v\), the transformed output becomes

Outside Changes Affect Outputs

In

the changes occur after \(f\) produces an output. They therefore alter the graph vertically.

• \(a\) scales or reflects output values;
• \(k\) translates output values vertically.

Inside Changes Affect Inputs

In

the changes occur before \(f\) receives its input. They therefore alter the graph horizontally.

• \(b\) scales or reflects input positions;
• \(h\) translates input positions horizontally.

Vertical Translation

Adding a constant outside the function gives

Every output increases by \(k\), so every point moves vertically:

Up and Down

If \(k>0\), the graph moves upward \(k\) units. If \(k<0\), the graph moves downward \(|k|\) units.

Example: Vertical Translation of a Parabola

Start with

Then

is the parent parabola translated upward \(5\) units. Its vertex moves from \((0,0)\) to

Horizontal Translation

Replacing \(x\) with \(x-h\) gives

Parent input \(u\) now occurs when \(x-h=u\), so

Therefore,

The Horizontal Sign Looks Reversed

The expression

moves the graph right \(4\) units, while

moves it left \(4\) units.

Why the Horizontal Sign Is Opposite

Suppose a parent feature occurs at input \(u\). In \(f(x-h)\), the same feature occurs when

Thus

The apparent sign reversal comes from solving an input equation, not from an arbitrary convention.

Example: Horizontal Translation of Absolute Value

The parent graph

has vertex \((0,0)\). The graph

is translated right \(3\) units, so its vertex is

Combining Horizontal and Vertical Translation

The form

translates the graph by the vector

Each point moves according to

Example: Two-Direction Translation

Starting from \(f(x)=x^2\), consider

The parent parabola moves right \(2\) and down \(7\). Its vertex moves to

Reflection Across the \(x\)-Axis

Multiplying the entire function by \(-1\) gives

Every output changes sign:

This is reflection across the \(x\)-axis.

Example: Vertical Reflection

The graph of

is the reflection of \(y=x^2\) across the \(x\)-axis. The upward-opening parabola becomes downward-opening.

Reflection Across the \(y\)-Axis

Replacing \(x\) with \(-x\) gives

Every input changes sign:

This reflects the graph across the \(y\)-axis.

Example: Horizontal Reflection

Let

Then

is the square-root graph reflected across the \(y\)-axis. It extends left from the origin instead of right.

Some Reflections May Be Invisible

If \(f\) is even, then

so reflection across the \(y\)-axis produces the same graph. For example,

Odd Functions and Reflections

If \(f\) is odd, then

For an odd function, reflecting across the \(y\)-axis produces the same result as reflecting across the \(x\)-axis.

Two Reflections Produce Origin Symmetry

Reflecting across both coordinate axes sends

This is equivalent to a \(180^\circ\) rotation about the origin.

Vertical Scaling

Multiplying the function by a nonzero constant \(a\) gives

Each output is multiplied by \(a\):

Vertical Stretch

If

vertical distances from the \(x\)-axis increase by the factor \(|a|\). This is a vertical stretch.

Example: Vertical Stretch

Compare

with

A parent point such as \((2,4)\) becomes

The parabola appears narrower because its outputs grow three times as far from the \(x\)-axis.

Vertical Compression

If

vertical distances from the \(x\)-axis shrink by the factor \(|a|\). This is a vertical compression.

Example: Vertical Compression

For

the point \((2,4)\) on \(y=x^2\) becomes

A Negative Vertical Scale Does Two Things

If \(a<0\), then

combines a vertical scale by \(|a|\) with a reflection across the \(x\)-axis.

For example,

reflects vertically and doubles all vertical distances.

Horizontal Scaling

Replacing \(x\) with \(bx\) gives

A parent input \(u\) appears at the new input \(u/b\), so

Horizontal Scaling Uses the Reciprocal

The coefficient inside the function acts reciprocally on horizontal distances.

multiplies horizontal coordinates by \(1/2\), while

multiplies horizontal coordinates by \(2\).

Horizontal Compression

If

then the horizontal scale factor is

The graph is compressed horizontally.

Example: Horizontal Compression

If \((4,7)\) lies on \(y=f(x)\), then on

the corresponding point is

Horizontal Stretch

If

then

so horizontal distances increase. This is a horizontal stretch.

Example: Horizontal Stretch

For

the point \((2,5)\) on the parent graph becomes

A Negative Horizontal Scale Also Reflects

If \(b<0\), then the horizontal coordinate mapping includes a sign change:

For the special form \(f(bx)\), this combines horizontal scaling with a reflection across the \(y\)-axis.

Example: Horizontal Scale and Reflection

If \((6,4)\) is on \(y=f(x)\), then on

the corresponding point is

Horizontal and Vertical Scaling Are Different

Compare

and

The first doubles outputs. The second halves horizontal coordinates. These transformations are generally not equivalent.

Special Coincidences Can Occur

For certain parent functions, two different-looking transformations may simplify to the same formula. For example, if \(f(x)=x^2\), then

This coincidence comes from the algebraic structure of the square function; it is not a general transformation rule.

The Full Transformation Form

Return to

Its effects are summarized by

• horizontal scale factor \(1/|b|\);
• horizontal reflection when \(b<0\);
• horizontal translation by \(h\);
• vertical scale factor \(|a|\);
• vertical reflection when \(a<0\);
• vertical translation by \(k\).

The Safest Point Rule

Rather than memorizing an order of verbal steps, transform a parent point directly:

This rule automatically handles signs, reciprocal horizontal scaling, and translations.

Example: Full Coordinate Transformation

Suppose \((6,-2)\) lies on \(y=f(x)\), and define

Here

Therefore,

The transformed point is

Reading \(h\) Requires Factored Inside Form

The horizontal translation is easiest to read when the inside expression is written as

Do not read \(h\) directly from an unfactored expression such as

Example: Factor Before Reading the Shift

Consider

Factor the inside expression:

Thus

The graph is horizontally compressed by \(1/2\) and translated right \(3\), not right \(6\).

Why \(f(2x-6)\) Is Not \(f(2(x-6))\)

These inputs are different:

while

Parentheses determine the actual horizontal shift.

Example: Transforming the Quadratic Parent

Let

Consider

Relative to the parent graph, this applies a vertical stretch by \(2\), a reflection across the \(x\)-axis, a translation right \(3\), and a translation up \(4\).

The Vertex Moves Predictably

The parent vertex \((0,0)\) transforms under

to

The negative coefficient changes the opening direction, but the vertex remains the translated image of the parent vertex.

Example: Transforming Absolute Value

Start with

Then

is vertically compressed by \(1/2\), translated left \(4\), and translated down \(3\). Its vertex is

Example: Transforming Square Root

Begin with

The function

moves the graph right \(5\), reflects it across the \(x\)-axis, and moves it up \(2\). The parent endpoint \((0,0)\) becomes

Example: Horizontal Scaling of Square Root

Consider

In transformation language, this is \(f(4x)\), a horizontal compression by \(1/4\).

Algebraically,

for real \(x\ge0\), so this particular parent function also makes the same graph look like a vertical stretch by \(2\).

Equivalent Formulas Can Describe the Same Graph

A single transformed graph may sometimes have several equivalent algebraic descriptions. Transformation language describes how one formula is built from a chosen parent function, but algebraic simplification may reveal an equivalent route.

Example: Cubic Transformation

Let

Then

reflects the cubic across the \(x\)-axis, translates it left \(2\), and translates it up \(1\). The parent center point \((0,0)\) moves to

Example: Reciprocal Transformation

From the parent

consider

This is

The graph is vertically stretched by \(2\), shifted right \(3\), and shifted down \(4\).

Distinguished Features Transform Too

Vertices, endpoints, centers, turning points, and other recognizable points move according to the same coordinate rule as every other point.

Identifying a few structural points often makes graphing much faster than constructing a large value table.

Transforming a Table of Points

Suppose a parent graph contains

For

the point rule is

Example: Transforming the Table

The three parent points become

The transformed points preserve the parent graph's structural relationship while changing location and vertical scale.

Transformations and the Domain

Vertical transformations do not change which input values are allowed. Horizontal transformations generally do.

If the parent domain is a set \(D\), then under

the new domain is obtained by transforming parent input coordinates:

Example: Domain Transformation

The parent square-root function has domain

For

the graph shifts right \(5\), so the domain becomes

Horizontal Reflection Can Reverse a One-Sided Domain

For

the parent domain \([0,\infty)\) reflects across the \(y\)-axis. The new domain is

Transformations and the Range

Horizontal transformations do not change the set of parent output values by themselves. Vertical transformations generally do.

If the parent range is \(R\), then transformed output values have the form

Example: Range Transformation

The parent quadratic \(f(x)=x^2\) has range

For

the vertical reflection and translation produce the range

Transformations Preserve Open and Closed Features

A transformation moves existing graph points but does not turn an excluded point into an included point merely because it moved. An open endpoint or hole remains excluded after transformation; a closed endpoint remains included.

Transforming a Hole

Suppose a parent relation has a hole at

Under

the hole moves to

It remains a hole.

Translations Do Not Change Shape or Size

A pure translation changes every point by the same displacement. Distances and angles within the graph are preserved.

The graph is relocated, not resized.

Reflections Preserve Size

A reflection reverses orientation but preserves distances. The reflected graph is congruent to the original graph.

Stretches and Compressions Change Scale

Scaling transformations change selected coordinate distances. Vertical scaling changes distances from the \(x\)-axis; horizontal scaling changes distances measured along the \(x\)-direction.

Scaling Relative to a Translated Center

In the full coordinate map

horizontal coordinates are scaled before being positioned around \(x=h\), while vertical outputs are scaled before being positioned around \(y=k\). This is why a negative \(b\) in the full transformed form reverses horizontal orientation relative to the translated vertical reference line.

Order Matters When Transformations Are Composed

Performing two transformations in different orders can produce different graphs.

For example, starting with an output \(y\), stretching vertically by \(2\) and then shifting up \(3\) gives

Shifting first and then stretching gives

The Formula Records the Actual Composition

Rather than applying verbal operations in an assumed order, read the algebraic grouping carefully or use the coordinate mapping. Parentheses determine which operations occur together.

Example: Two Different Vertical Compositions

Compare

with

These functions differ by a vertical translation of \(3\) units. The parentheses matter.

Inside Composition Needs Equal Care

Compare

with

The first contains the horizontal shift \(h=3\). The second rewrites as

so its shift is \(3/2\).

Transformations Can Be Reversed

If \(a\ne0\) and \(b\ne0\), the coordinate transformation

is reversible. From a transformed point \((x,y)\), recover the parent coordinates by

Why \(a\) and \(b\) Are Usually Nonzero

In the standard transformation model, \(a\ne0\) and \(b\ne0\) are assumed when we want a reversible reshaping of the parent graph.

If \(a=0\), then

and the parent output structure collapses to a horizontal line.

What Happens If \(b=0\)?

If

and \(f(0)\) exists, the output is constant for every allowed \(x\). This is not an ordinary horizontal stretch; the parent input variation has collapsed to one value.

Transforming Relations, Not Only Functions

Coordinate transformations can also be applied to graphs that are relations but not functions. Each point moves according to the specified coordinate rule.

For example, reflecting a circle across the \(y\)-axis still produces a circle, even though the full circle is not a function of \(x\).

Translation of an Implicit Relation

The circle

translated right \(4\) and up \(2\) becomes

The center moves from \((0,0)\) to \((4,2)\).

Transformations and Symmetry

A transformation may move a graph's symmetry away from the coordinate axes. For example, \(y=x^2\) is symmetric about the \(y\)-axis, but

is symmetric about

Family Symmetry Can Move with the Graph

A transformed graph can preserve the structural symmetry of its parent family without remaining even or odd relative to the origin. Coordinate-axis symmetry and family structure are related but not identical ideas.

Graph Shape Versus Graph Position

Translations primarily change position. Reflections change orientation. Stretches and compressions change scale. These distinctions help describe what changed without confusing the graph's family with its location.

Recognizing Transformations from a Graph

When comparing a transformed graph with a known parent, look for structural landmarks such as:

• vertices;
• endpoints;
• centers;
• turning points;
• symmetry lines;
• distances between recognizable points.

Use More Than One Point to Detect Scale

A translation can often be inferred from one distinguished point. A scale factor usually requires comparison of additional points or distances.

Several transformed graphs can share the same vertex but have different vertical or horizontal scales.

Example: Same Vertex, Different Scale

All three functions

have vertex \((2,1)\), but their vertical scales differ.

Do Not Use Visual Width Alone

A graph may look narrow or wide because of axis scale, window dimensions, or aspect ratio. Identify a stretch or compression by coordinates and algebraic structure, not merely by appearance on a screen.

Graphing Windows Can Distort Perception

If one grid unit horizontally occupies a different physical length from one grid unit vertically, angles and apparent steepness can be distorted. The transformation should be interpreted using coordinate values rather than the physical shape of the display alone.

Transformation Language Is Relative

A graph is transformed relative to a chosen reference graph. The same final function may be described in different ways if a different reference function is chosen.

Example: Choosing the Reference Matters

The function

is a vertical stretch by \(4\) of \(f(x)=x^2\). It can also be viewed as a horizontal compression by \(1/2\) because

Both descriptions are valid relative to the same quadratic parent, but they arise from different equivalent algebraic forms.

Equivalent Descriptions Are Not Always Available

For a general function \(f\), there is no reason to expect

Transformation equivalences that work for one family should not be assumed for another.

Transformations and Zeros

Suppose \(u\) is a zero of the parent function, so \(f(u)=0\). Under a purely horizontal transformation

that zero moves to

Vertical Translation Can Change Zeros

If \(k\ne0\), a parent zero does not generally remain an \(x\)-intercept, because the transformed output is

which equals \(k\) when \(v=0\).

Transformations and the \(y\)-Intercept

The transformed \(y\)-intercept must be evaluated at \(x=0\):

It does not generally come from simply translating the parent \(y\)-intercept, because horizontal transformations may move that parent point away from the \(y\)-axis.

Example: A Parent \(y\)-Intercept Moves Off the Axis

The parent parabola \(y=x^2\) has \(y\)-intercept \((0,0)\). Under

that particular parent point moves to \((3,0)\), which is no longer on the \(y\)-axis. The new \(y\)-intercept is instead

Transformations and Extreme Points

A local or absolute extreme point of a parent graph moves with the coordinate transformation, provided the transformation is reversible. A negative vertical scale reverses high and low behavior, so maxima and minima exchange roles.

Example: Minimum Becomes Maximum

The parent parabola \(y=x^2\) has an absolute minimum at \((0,0)\). The graph

reflects vertically and shifts upward. The transformed point \((0,5)\) is an absolute maximum.

Transformations of Increasing and Decreasing Behavior

A negative vertical scale reverses increasing and decreasing behavior. A negative horizontal scale also reverses left-to-right orientation. Positive scale factors preserve orientation in their respective directions.

Transforming Piecewise Graphs

The same transformation rule applies to every point of a piecewise graph. Endpoints, open circles, closed circles, and each piece move together.

However, domain conditions written algebraically must also be transformed consistently.

Example: Moving an Endpoint

Suppose a parent piece begins at the included endpoint \((1,2)\). Under

the coordinate rule gives

The endpoint remains included.

Transformations of Discrete Graphs

Discrete points transform exactly as continuous graphs do. If the original relation consists only of selected points, the transformed relation also consists only of their images. Do not connect them unless the context says to do so.

Example: Transforming a Discrete Point

Under

we can write

Thus a parent point \((3,5)\) maps to

A Reliable Procedure: Reading a Transformation

1. Identify the parent function \(f\).
2. Rewrite the formula in the form \(a f(b(x-h))+k\) when possible.
3. Factor the inside expression before reading \(b\) and \(h\).
4. Record \(a\), \(b\), \(h\), and \(k\).
5. Use horizontal scale factor \(1/|b|\).
6. Use vertical scale factor \(|a|\).
7. Check signs for horizontal or vertical reflections.
8. Apply translations by \(h\) and \(k\).
9. Verify important points using the coordinate rule.

A Reliable Procedure: Transforming Known Points

1. Choose useful parent points \((u,v)\).
2. Compute \(x=h+u/b\).
3. Compute \(y=k+av\).
4. Plot the transformed points.
5. Preserve open or closed endpoint status.
6. Connect points only when the parent relation is continuous there.
7. Use family structure to complete the graph.

A Reliable Procedure: Recovering the Parent Point

If a point \((x,y)\) is known on the transformed graph and \(a,b\ne0\), use

Then \((u,v)\) is the corresponding point on the parent graph.

Common Error: Reversing the Vertical Shift

In

the \(+5\) moves the graph upward, not downward. Outside additions act directly on outputs.

Common Error: Reading the Horizontal Shift Directly

In

the graph moves right \(5\), not left \(5\). Solve \(x-5=u\) to see why.

Common Error: Forgetting the Reciprocal Horizontal Scale

The graph of

is horizontally compressed by a factor of \(1/4\), not stretched by a factor of \(4\).

Common Error: Treating \(2f(x)\) and \(f(2x)\) as the Same

The first scales outputs; the second rescales inputs. They are different transformations unless the particular function happens to make the formulas equivalent.

Common Error: Ignoring a Negative Inside Coefficient

In

the magnitude \(2\) causes horizontal compression, while the negative sign causes horizontal reflection.

Common Error: Ignoring a Negative Outside Coefficient

In

the magnitude \(3\) produces a vertical stretch, while the negative sign reflects the graph across the \(x\)-axis.

Common Error: Reading \(h\) Before Factoring

From

first write

The horizontal shift is left \(4\), not left \(12\).

Common Error: Moving Only a Special Point

A transformation acts on the entire graph. Moving the vertex correctly but leaving other points unchanged does not produce the transformed function.

Common Error: Connecting Discrete Images

Transforming a discrete relation relocates its points; it does not make the relation continuous. Preserve the original connectivity structure.

Common Error: Losing Endpoint Status

An open circle represents an excluded point. A geometric transformation moves that excluded location but does not fill it in. Likewise, an included endpoint remains included.

Common Error: Trusting Appearance Over Coordinates

A graphing window can make a vertical stretch appear like a horizontal compression or vice versa. Use numerical scale, known points, and the formula to identify transformations precisely.

Common Error: Assuming Every Formula Is Already in Transformation Form

Expressions often need algebraic rewriting before their transformations are clear. Factor inside coefficients and identify the parent function before assigning transformation parameters.

Checking a Transformation Algebraically

After describing a transformation, substitute the parent formula into the transformation expression and simplify. The result should match the proposed transformed formula.

Example: Algebraic Check

Let \(f(x)=x^2\). A vertical stretch by \(3\), shift right \(2\), and shift down \(1\) gives

Substituting the parent rule gives

Checking a Transformation with a Point

If \((1,1)\) lies on \(y=x^2\), then under

the coordinate rule predicts

Check directly:

Checking Several Points Is Stronger

One matching point can occur by accident. Checking several structural points provides stronger evidence that the intended transformation has been read correctly.

Transformation Notation as a Toolbox

The general form

is not a new function family. It is a reusable language for describing how a known graph changes.

One Transformation at a Time

When a complicated formula feels difficult, isolate the parameters and study one effect at a time. For example, from \(f(x)\) compare successively with

Each step adds one visible change.

Then Recombine with the Coordinate Rule

Once individual effects are understood, the point mapping

provides a compact way to apply them together without losing track of signs or scale factors.

Practice 1: Vertical Translation

Describe the transformation from \(y=f(x)\) to

Practice 2: Horizontal Translation

Describe the transformation from \(y=f(x)\) to

Practice 3: Vertical Reflection and Stretch

Describe the transformation represented by

Practice 4: Horizontal Scale

Describe the horizontal scaling in

Practice 5: Transform a Point

Suppose \((6,3)\) lies on \(y=f(x)\). Find the corresponding point on

Practice 6: Read the Inside Correctly

Rewrite

in the form \(f(b(x-h))\), then identify the horizontal scale and translation.

Practice 7: Quadratic Parent

Starting from \(f(x)=x^2\), describe all transformations in

Practice 8: Square-Root Endpoint

The parent graph \(y=\sqrt{x}\) begins at \((0,0)\). Where does this endpoint move under

Practice 9: Recover a Parent Point

The point \((7,-5)\) lies on

Find the corresponding point on \(y=f(x)\).

Practice 10: Identify the Error

A student says that

is the graph of \(f\) compressed horizontally by \(1/2\) and shifted right \(8\). Explain the error and give the correct translation.

Practice Answers

Practice 1

The graph is translated down \(7\) units:

Practice 2

Since

the graph is translated left \(5\) units.

Practice 3

The factor \(-4\) causes a vertical stretch by \(4\) and a reflection across the \(x\)-axis.

Practice 4

The horizontal scale factor is the reciprocal:

Thus the graph is horizontally compressed by a factor of \(1/5\).

Practice 5

Here

Transform \((u,v)=(6,3)\):

Therefore,

Practice 6

Therefore,

The graph is horizontally compressed by \(1/4\) and translated left \(3\).

Practice 7

Relative to \(y=x^2\), the graph is vertically compressed by \(1/2\), reflected across the \(x\)-axis, translated right \(3\), and translated up \(8\). Its vertex is

Practice 8

The expression \(x+2=x-(-2)\) shifts the endpoint left \(2\). The vertical factor and translation send \(0\) to

Thus the endpoint moves to

Practice 9

Use the inverse coordinate rule:

The parent point is

Practice 10

Factor the inside expression:

The horizontal compression by \(1/2\) is correct, but the translation is right \(4\), not right \(8\):

What to Remember

For

• \(h\) translates horizontally;
• \(k\) translates vertically;
• \(|a|\) is the vertical scale factor;
• \(a<0\) reflects vertically;
• \(1/|b|\) is the horizontal scale factor;
• \(b<0\) reverses horizontal orientation;
• inside coefficients act reciprocally on horizontal coordinates;
• factor the inside expression before reading the horizontal shift;
• open, closed, continuous, and discrete features transform with the graph.

The most reliable point rule is

Where We Go Next

Transformations explain how a familiar graph moves, flips, and changes scale. Many graphs, however, are built from separate rules or contain breaks, jumps, holes, or other interruptions that require additional graphical language.

The next lesson, Piecewise Graphs and Discontinuities, examines graphs defined by different formulas on different parts of the domain and develops methods for interpreting endpoints, jumps, holes, and other forms of discontinuous behavior.