Transformation of Functions

13 min read

Quick Reference

Try This First

Before any formula, take the points of one small graph and move them by hand.

Suppose a function $f$ passes through these three points:

Now build a brand new table for $g(x) = f(x) + 3$. You do not need a rule for $f$. You only need to add $3$ to each output.

Hold onto one observation from that table: a change made to the output (adding $3$ after $f$ does its work) moved the graph vertically. A change made to the input (which you have not tried yet) will move the graph horizontally, and it will behave in the opposite direction from what most people expect. That opposite-direction surprise is the central idea to watch for.

An Intuitive Introduction

A function is a process that takes an input and gives back an output. A transformation is a small instruction added either before that process runs (a change to the input) or after it finishes (a change to the output). Each instruction reshapes the graph in a predictable way.

There are three families of instruction, and each comes in an outside version (acting on the output) and an inside version (acting on the input):

• Shifts slide the graph without changing its shape. Add a constant outside to move up or down; add a constant inside to move left or right.
• Reflections flip the graph across an axis. A minus sign outside flips it across the $x$-axis; a minus sign inside flips it across the $y$-axis.
• Stretches and compressions scale the graph. A factor outside scales heights (vertical); a factor inside scales widths (horizontal).

The single rule that organizes all of this: changes on the outside of $f$ do exactly what they look like they do to the output, and changes on the inside of $f$ do the opposite of what they look like to the input. Replacing $x$ with $x + 2$ moves the graph two units to the left, not right. The worked examples and misconceptions below return to this point because it is where most errors begin.

Prerequisite Hub

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    subgraph Builds_On["Builds on"]
        A["Transformations<br/>of Graphs"]
        B["Functions and<br/>Function Notation"]
    end

    subgraph ThisSkill["This Skill"]
        C["Transformation<br/>of Functions"]
    end

    subgraph Unlocks
        D["Absolute Value<br/>Functions"]
        E["Inverse<br/>Functions"]
    end

    A --> C
    B --> C
    C --> D
    C --> E

    style C fill:#d1fae5,stroke:#a565f0,stroke-width:3px

Builds on (master these first):

Unlocks (these come next):

There are no cross-course prerequisites for this skill.

Before You Start: Prerequisite Self-Check

Official Definitions

These statements are quoted faithfully from the source. Read them slowly. Each one names a transformation and tells you exactly what to look for.

Vertical reflection. Given a function $f(x)$, a new function $g(x) = -f(x)$ is a vertical reflection of the function $f(x)$, sometimes called a reflection about (or over, or through) the $x$-axis.

Source: OpenStax Precalculus 2e, Section 1.5 Transformation of Functions.

Vertical stretch or compression. Given a function $f(x)$, a new function $g(x) = a \cdot f(x)$, where $a$ is a constant, is a vertical stretch or compression. If $a > 1$ the graph is stretched; if $0 < a < 1$ it is compressed; if $a < 0$ it combines a stretch or compression with a vertical reflection.

Source: OpenStax Precalculus 2e, Section 1.5 Transformation of Functions.

Even and odd functions. A function is even if for every input $x$, $f(x) = f(-x)$, and its graph is symmetric about the $y$-axis. A function is odd if for every input $x$, $f(-x) = -f(x)$, and its graph is symmetric about the origin.

Source: OpenStax Precalculus 2e, Section 1.5 Transformation of Functions.

The source for Section 1.5 does not state a separate named theorem. The definitions above carry the section, together with the inside-versus-outside rule developed in the worked examples.

Two Representations Side by Side

Read the same transformation three ways at once: as an instruction, as algebra, and as an effect on a sample point.

The column on the right is the one to translate back to a graph. Pick any row, take the point $(2, 4)$ from the Try This First table, and apply the arrow. Then ask yourself: does the new point sit where my mental picture of that shift would put it?

Worked Examples

Each example asks you to predict first, then compute, then compare. The prediction is where the learning happens.

Example 1: A vertical shift

Let $f(x) = x^2$. Describe and graph $g(x) = f(x) - 2$.

Predict. The change is outside $f$, so it acts on the output. Subtracting $2$ should pull every height down by $2$. Predict that the parabola slides down $2$ units, with its lowest point moving from $(0, 0)$ to $(0, -2)$.

Compute. Build a table from the rule $g(x) = x^2 - 2$.

Compare. Every output dropped by exactly $2$, and the vertex moved to $(0, -2)$. The prediction holds. The shape of the parabola did not change, only its vertical position.

Example 2: A horizontal shift (the surprising direction)

Let $f(x) = x^2$. Describe and graph $g(x) = f(x + 3)$.

Predict. The change is inside $f$, so it acts on the input and runs opposite to its sign. The $+3$ inside should move the graph $3$ units to the left, not right. Predict the vertex moves from $(0, 0)$ to $(-3, 0)$.

Compute. The rule is $g(x) = (x + 3)^2$. Ask which input makes the inside equal to $0$, since $0$ is where the original vertex sat. Setting $x + 3 = 0$ gives $x = -3$.

Compare. The vertex landed at $(-3, 0)$, exactly $3$ units left. The prediction holds. The reason is concrete: to feed $f$ the input $0$, you must now choose $x = -3$, because the function adds $3$ before squaring. The graph shifts toward the negative direction to compensate.

Example 3: A vertical reflection

Let $f(x) = \sqrt{x}$, whose graph starts at the origin and rises to the right. Describe $g(x) = -f(x)$.

Predict. A minus sign outside flips outputs to their opposites. Predict the curve flips across the $x$-axis: where $f$ was above the axis, $g$ is the same distance below.

Compute. Using $g(x) = -\sqrt{x}$:

Compare. Each point $(x, y)$ became $(x, -y)$, matching the official definition of a vertical reflection: $g(x) = -f(x)$ reflects $f$ about the $x$-axis. The prediction holds.

Example 4: A vertical stretch, then reflection

Let $f(x) = \lvert x \rvert$. Describe $g(x) = -2\,f(x)$.

Predict. Compare the constant $a = -2$ to the official definition. Since $\lvert a \rvert = 2 > 1$, the heights stretch by a factor of $2$. Since $a < 0$, the stretch is combined with a vertical reflection. Predict a V shape that opens downward and is twice as steep as $y = \lvert x \rvert$.

Compute. Using $g(x) = -2\lvert x \rvert$:

Compare. Each height doubled (the stretch by $a = 2$ in size) and then flipped sign (the reflection from $a < 0$). The vertex stayed at $(0, 0)$ because $2 \cdot 0 = 0$. The prediction holds, and it lines up exactly with the official definition: $a < 0$ combines a stretch with a vertical reflection.

Example 5: Testing even or odd from the definition

Decide whether $f(x) = x^4 - 3x^2$ is even, odd, or neither.

Predict. The graph is built only from even powers of $x$, so a guess based on the picture is that it is symmetric about the $y$-axis, that is, even. Now confirm it from the definition rather than from the picture alone.

Compute. Replace $x$ with $-x$ throughout:

\[ f(-x) = (-x)^4 - 3(-x)^2 = x^4 - 3x^2. \]

Compare $f(-x)$ to $f(x)$ and to $-f(x)$:

• $f(-x) = x^4 - 3x^2 = f(x)$, so the even condition $f(x) = f(-x)$ holds for every input $x$.
• $-f(x) = -x^4 + 3x^2$, which is not equal to $f(-x)$, so the odd condition fails.

Compare. The definition confirms the prediction: $f$ is even, and its graph is symmetric about the $y$-axis. Notice that the conclusion came from the algebra, $f(x) = f(-x)$ for all $x$, not from trusting the shape of the picture.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can describe a vertical shift from a rule of the form $f(x) + k$
• I can determine the direction of a horizontal shift from $f(x + h)$ or $f(x - h)$, including the opposite-direction surprise
• I can translate a transformation into an effect on a sample point $(x, y)$

Competent (Level 3-4):

• I can classify a function as even, odd, or neither using $f(x) = f(-x)$ and $f(-x) = -f(x)$
• I can identify a vertical reflection as $g(x) = -f(x)$ and a vertical stretch or compression as $g(x) = a\,f(x)$
• I can combine a stretch and a reflection into a single constant $a$ and read its meaning from the official definition

Proficient (Level 5):

• I can apply two transformations in a stated order and write the resulting rule
• I can explain why the order of a reflection and a shift changes the result
• I can justify the direction of an inside change by tracking which input reaches a given stage of the process

Mental Model

Picture the function as a small machine on an assembly line. The input $x$ goes in one end, the process runs, and the output comes out the other end.

• A change you bolt onto the output end (outside $f$) does exactly what it says: add $3$ and every result is $3$ higher; multiply by $-1$ and every result flips sign.
• A change you bolt onto the input end (inside $f$) adjusts what reaches the machine, so it appears to run backward from your point of view: to feed the machine the input it used to get from $x$, you now have to supply a different $x$, which shifts the graph the opposite way.

Outside is honest; inside is contrary. That one sentence predicts the direction of every shift, reflection, and scaling in this section.

Connections

Looking back:

• Transformations of Graphs supplies the graph-reading and graph-sketching this skill depends on.
• Functions and Function Notation supplies the notation $f(x + h)$, $a\,f(x)$, and the input-output process view used throughout.

Looking ahead:

• Absolute Value Functions reuse shifts, stretches, and reflections applied to $y = \lvert x \rvert$.
• Inverse Functions build on the reflection idea, this time across the line $y = x$.

Real-world connections:

• Converting a temperature scale (multiply, then add a constant) is a vertical stretch followed by a vertical shift.
• Adjusting the volume of a recorded sound scales the output of a fixed waveform, a vertical stretch.
• Delaying a signal by a fixed time shifts its graph horizontally, the inside change applied to the time input.

Resources

The page range for Section 1.5 is pending faculty verification and will be added once confirmed against the print edition.

Last updated: 2026-06-16