Transformations of Graphs
Quick Reference
| Primary source | OpenStax Precalculus 2e |
| Chapter | Chapter 1: Functions |
| Section | Section 1.5: Transformation of Functions |
| Subsection | 1.5 Transformation of Functions |
| Pages | Page numbers pending faculty verification. |
| Course | MATH141 (Precalculus) |
| Standing | Week 0 review skill; bridges basic graphing to absolute-value and inverse functions |
Lecture pointer: this skill is the place to make the function-as-process view explicit. A transformation does not move the curve as a picture; it rewrites the input-output rule, and the picture follows from that. Open class with one parent graph and ask students to predict the new graph before any rule is written.
Before You Start: Prerequisite Check
📋 Can you do these? (Click to reveal self-test)
Test yourself on these prerequisite skills:
Reading a graph value: The graph of $f$ passes through the point $(2, 5)$. What is $f(2)$?
Check
$f(2) = 5$. The output at input $2$ is $5$.
Plotting from a rule: If $f(x) = x^2$, list the points for $x = -2, -1, 0, 1, 2$.
Check
$(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 4)$.
Substituting into a function: If $f(x) = \sqrt{x}$, what is $f(x - 3)$ as a rule?
Check
$f(x - 3) = \sqrt{x - 3}$. Replace every $x$ in the rule with $x - 3$.
If you struggled:
- Review Graphing Functions for plotting from a table or rule and reading a value off a graph.
Try This First
Before any rule, take three minutes with a parent graph and a pencil.
Start with the graph of $f(x) = x^2$, the parabola with its low point at the origin. Now sketch, on the same axes, the points you would get if every output went up by $3$. The point $(0, 0)$ becomes $(0, 3)$. The point $(2, 4)$ becomes $(2, 7)$. The point $(-1, 1)$ becomes $(-1, 4)$.
Notice what happened to the whole curve, and predict in words where the low point landed and which direction the parabola moved. Then read on and check your prediction against the vertical-shift rule. There is no need to rush this; building the picture by hand first is what makes the symbols make sense later.
What a Transformation Is Doing
A function is a process that takes an input and gives back an output. A transformation is a small edit to that process, made before or after the function runs, and the graph changes to match.
Two questions organize every transformation in this skill:
- Did the edit happen to the output (the value $f(x)$ produces) or to the input (the $x$ that goes in)?
- An output edit moves the graph vertically; an input edit moves the graph horizontally.
Output edits behave the way intuition expects. Adding $3$ to the output raises the graph by $3$. Input edits run the opposite way to most first guesses, and that surprise is exactly where the careful work in this skill lives.
A useful frame: each point $(a, b)$ on the original graph is sent to a new point on the transformed graph. Track one point, and the rule for the whole curve follows.
Prerequisite Hub
Builds on:
| Skill | Why it is needed |
|---|---|
Graphing Functions (graphing-functions-basic) |
A transformation is read by comparing a new graph to the parent graph, so plotting from a rule and reading a value off a graph come first. |
Unlocks:
| Skill | What this skill gives it |
|---|---|
Absolute Value Functions (m141-absolute-value-functions) |
The graph of $y = \lvert x \rvert$ is studied by shifting, stretching, and reflecting the V-shaped parent graph. |
Inverse Functions (m141-inverse-functions) |
The graph of an inverse is the original graph reflected across the line $y = x$, a reflection in the same family studied here. |
Quick Reference
| Property | Value |
|---|---|
| Chapter | 1.5 |
| Course | MATH141 |
| Difficulty | Foundational |
| Time | ~40 minutes |
Transformations at a Glance
| New rule | Edit acts on | Effect on the graph | Direction rule |
|---|---|---|---|
| $g(x) = f(x) + k$ | output | vertical shift | up if $k > 0$, down if $k < 0$ |
| $g(x) = f(x - h)$ | input | horizontal shift | right if $h > 0$, left if $h < 0$ |
Two connected representations of the same vertical shift $g(x) = f(x) + 3$ with $f(x) = x^2$:
| $x$ | $f(x) = x^2$ | $g(x) = f(x) + 3$ |
|---|---|---|
| $-2$ | $4$ | $7$ |
| $-1$ | $1$ | $4$ |
| $0$ | $0$ | $3$ |
| $1$ | $1$ | $4$ |
| $2$ | $4$ | $7$ |
Every entry in the third column is the matching entry in the second column plus $3$. On the graph, every point sits exactly $3$ units higher. Read across the table, then read the same fact off the graph, and confirm they say the same thing.
Official Definitions
Vertical Shift
Given a function $f(x)$, a new function $g(x) = f(x) + k$, where $k$ is a constant, is a vertical shift of the function $f(x)$. All the output values change by $k$ units. If $k$ is positive, the graph will shift up. If $k$ is negative, the graph will shift down.
Source: OpenStax Precalculus 2e, Section 1.5 Transformation of Functions.
Horizontal Shift
Given a function $f$, a new function $g(x) = f(x - h)$, where $h$ is a constant, is a horizontal shift of the function $f$. If $h$ is positive, the graph will shift right. If $h$ is negative, the graph will shift left.
Source: OpenStax Precalculus 2e, Section 1.5 Transformation of Functions.
Reading the Two Rules Together
| Form | Edit point | Where the constant lives | Behaves as expected? |
|---|---|---|---|
| $g(x) = f(x) + k$ | output, after $f$ runs | added outside $f$ | yes: $+k$ moves up |
| $g(x) = f(x - h)$ | input, before $f$ runs | subtracted inside $f$ | no: $-h$ moves right |
The horizontal case is written with a minus sign on purpose. In $g(x) = f(x - h)$, a positive $h$ moves the graph to the right, because the input must be $h$ units larger to reach the same output the parent graph had. This sign flip is the single most missed idea in the section, and it is worth slowing down for.
Worked Examples
Worked Example 1: Vertical Shift (Output Edit)
Let $f(x) = x^2$. Describe and graph $g(x) = f(x) + 4$.
Predict first. The edit is on the output ($+4$ is added after $f$ runs), and $4 > 0$, so predict the parabola moves up by $4$, with its low point rising from $(0, 0)$ to $(0, 4)$.
Now check by tracking points.
| $x$ | $f(x) = x^2$ | $g(x) = x^2 + 4$ |
|---|---|---|
| $-2$ | $4$ | $8$ |
| $0$ | $0$ | $4$ |
| $2$ | $4$ | $8$ |
Each output of $g$ is the matching output of $f$ plus $4$. The low point moved from $(0, 0)$ to $(0, 4)$.
Compare to the prediction. The graph rose by $4$ and the low point landed at $(0, 4)$, exactly as predicted. By the vertical-shift definition with $k = 4 > 0$, the graph of $g$ is the graph of $f$ shifted up $4$ units.
Worked Example 2: Horizontal Shift (Input Edit)
Let $f(x) = \sqrt{x}$, whose graph starts at the origin and rises to the right. Describe and graph $g(x) = f(x - 3)$.
Predict first. The edit is on the input, with $g(x) = f(x - 3)$, so $h = 3$. The definition says a positive $h$ shifts the graph right, so predict the starting point moves from $(0, 0)$ to $(3, 0)$.
Now check by tracking points. The parent graph passes through $(0, 0)$, $(1, 1)$, and $(4, 2)$. For $g$, the rule is $g(x) = \sqrt{x - 3}$. To get the same output the parent had at input $a$, the input to $g$ must be $a + 3$.
| Parent point on $f$ | Matching point on $g$ |
|---|---|
| $(0, 0)$ | $(3, 0)$ |
| $(1, 1)$ | $(4, 1)$ |
| $(4, 2)$ | $(7, 2)$ |
Check one row directly: $g(4) = \sqrt{4 - 3} = \sqrt{1} = 1$, which matches the point $(4, 1)$.
Compare to the prediction. The starting point moved from $(0, 0)$ to $(3, 0)$, a shift of $3$ units to the right, exactly as the definition with $h = 3 > 0$ predicts. The minus sign inside $f$ produced a rightward move, not a leftward one.
Worked Example 3: Reading a Shift from Two Graphs
Suppose the graph of $g$ looks exactly like the graph of $f$ but every point is $2$ units lower and $5$ units to the left. Write $g$ in terms of $f$.
Predict first. Lower means a negative output edit, so expect $+k$ with $k = -2$. Left means a negative horizontal shift, so expect $f(x - h)$ with $h = -5$.
Now build the rule. Down $2$ units is the output edit $f(x) - 2$. Left $5$ units is the input edit $f(x - (-5)) = f(x + 5)$. Combining, $$g(x) = f(x + 5) - 2.$$
Compare to the prediction. The constant outside is $-2$ (a downward shift, matching “$2$ units lower”), and inside the function the input is $x + 5$, which is $x - h$ with $h = -5$ (a leftward shift, matching “$5$ units to the left”). Both pieces agree with the read of the picture.
Common Misconceptions
the graph is a literal picture that you slide, so the algebra should match the eye. A graph is the record of an input-output process, not a fixed picture to drag. Reading it as a literal object is why the horizontal direction feels backward. In $g(x) = f(x - 3)$, the minus sign suggests “left” to the eye, but the graph moves right. The reason is in the process: to make $g$ give the same output the parent gave at input $0$, the input to $g$ must be $3$, because $g(3) = f(3 - 3) = f(0)$. The new graph reaches each old output $3$ units later along the $x$-axis, which is a shift to the right. Track one point through the rule and the direction stops being a guess.
a constant added inside the function moves the graph vertically. Where the constant sits decides everything. A constant added or subtracted outside $f$, as in $f(x) + k$, edits the output and moves the graph vertically. A constant subtracted inside $f$, as in $f(x - h)$, edits the input and moves the graph horizontally. Test it on $f(x) = x^2$: the rule $f(x) + 3 = x^2 + 3$ raises the low point to $(0, 3)$, while $f(x - 3) = (x - 3)^2$ slides the low point to $(3, 0)$. Same number, completely different move, decided only by inside versus outside.
in $f(x - h)$, the sign of $h$ tells you the direction directly. The shift goes opposite to the sign that appears next to $x$. The form is written as $f(x - h)$, so $h$ itself is positive for a rightward shift. If the rule shows $f(x + 5)$, rewrite it as $f(x - (-5))$ to read off $h = -5$, which is a shift to the left. Matching the rule to the exact form $f(x - h)$ before reading the direction prevents the sign slip.
Practice Problems
The graph of $g(x) = f(x) - 6$ is obtained from the graph of $f$ by which transformation?
The graph of $g(x) = f(x - 8)$ is obtained from the graph of $f$ by which transformation?
The function $f$ has $f(0) = 2$, $f(1) = 5$, and $f(2) = 4$. Let $g(x) = f(x) + 3$. Find $g(0)$, $g(1)$, and $g(2)$.
Let $f(x) = x^2$ and $g(x) = f(x - 2)$. Write $g$ as an explicit rule in $x$, then find $g(5)$. Where did the low point of the parabola move?
The graph of $g$ is the graph of $f$ shifted right $4$ units and down $1$ unit. Write $g$ in terms of $f$. If $f(x) = \sqrt{x}$, write $g$ as an explicit rule and state the point where its graph begins.
Why the Horizontal Direction Reverses
🔍 One way to convince a classmate (and one more way to see it)
A classmate insists that $f(x - 3)$ should move the graph left because of the minus sign. Here are two ways to settle it.
One way (track a single point). Pick the input where the parent graph does something memorable. For $f(x) = \sqrt{x}$, the memorable point is the start at $x = 0$. The new graph $g(x) = f(x - 3)$ does that same memorable thing when its inside expression equals $0$, that is, when $x - 3 = 0$, so $x = 3$. The start moved from $x = 0$ to $x = 3$, which is to the right.
Another way (read it as a question about input). Ask what input $g$ needs to match a given output of $f$. Since $g(x) = f(x - 3)$, the input to $g$ must be $3$ larger than the input $f$ used, to feed $f$ the same number. Needing a larger input means reaching the value later along the $x$-axis, which is a rightward shift.
Which explanation is more convincing to you, and why? Either one is valid; the point is that the direction comes from the process, not from the look of the sign.
Mental Model
The “edit the recipe” analogy:
Think of $f$ as a recipe that turns an input into an output. A vertical shift edits the result after the recipe finishes: cook as usual, then add $k$ to whatever came out. That is why $+k$ raises the graph in the obvious direction.
A horizontal shift edits the input before the recipe runs: change the number you feed in, then cook. Because you are altering the ingredient list rather than the finished dish, the effect on the graph runs opposite to first intuition. To land on the same output the parent recipe produced, you have to feed in a number that is $h$ larger, which pushes the matching point to the right.
Outside the function edits the answer; inside the function edits the question.
Mastery Checklist
Novice (Level 1-2):
Competent (Level 3-4):
Proficient (Level 5):
Connections
Looking back:
- Graphing Functions supplies the plotting and point-reading skills a transformation is built on.
Looking ahead:
- Absolute Value Functions (
m141-absolute-value-functions): the V-shaped parent graph $y = \lvert x \rvert$ is shifted and reflected using exactly these rules. - Inverse Functions (
m141-inverse-functions): an inverse graph is the original reflected across the line $y = x$, a reflection in the same transformation family.
Real-world connections:
- A temperature graph reported in a different baseline (sea level versus a mountain station) is a vertical shift of the same daily pattern.
- A delayed start to a repeating signal (a sound that begins three seconds later) is a horizontal shift of the original waveform.
Resources
| Resource | Link |
|---|---|
| OpenStax Precalculus 2e, Section 1.5: Transformation of Functions | https://openstax.org/books/precalculus-2e/pages/1-5-transformation-of-functions |
| OpenStax Precalculus 2e, Chapter 1 Key Concepts | https://openstax.org/books/precalculus-2e/pages/1-key-concepts |
| Previous | Up | Next |
|---|---|---|
| Graphing Functions | Skills Index | Absolute Value Functions |
Last updated: 2026-06-16