Transformation of Functions: Shifts, Flips & Stretches

Definitionnoun phrase

What is a transformation of a function?

Transformation of functions is changing a function’s rule so that its graph moves, flips or resizes. Adding k outside, f(x) + k, shifts the graph up; replacing x with x − h shifts it right; −f(x) reflects it over the x-axis; and a · f(x) stretches it vertically. The basic graph being changed is the parent function.

In Spanish: transformación de funciones
Also called: function transformations, graph transformations, shifts, reflections and stretches, transforming functions
Grade level: Algebra 2

Algebra 1Algebra 2PrecalculusCCSS HSF-BF.B.3SATACTPSATACCUPLACER

Transformation of Functions in a picture

Each green graph is the dashed parent y = |x| after one change. Note the top right: f(x − 3) moves the graph right, even though there is a minus sign.

Formal definition and notation

Given a function f, the functions f(x) + k, f(x − h), a · f(x) and f(bx) are transformations of f. f(x) + k translates the graph k units vertically; f(x − h) translates it h units horizontally; a · f(x) stretches it vertically by |a|, reflecting it across the x-axis if a < 0; f(bx) compresses it horizontally by the factor |b|, reflecting it across the y-axis if b < 0. A point (x, y) on f maps to ( xb + h, ay + k ) on y = a · f(b(x − h)) + k.

Symbol	Read it as	Meaning
f(x) + k	f of x plus k	vertical shift: up k (down if k < 0)
f(x − h)	f of x minus h	horizontal shift: right h (left if h < 0)
−f(x)	negative f of x	reflection across the x-axis
f(−x)	f of negative x	reflection across the y-axis
a · f(x)	a times f of x	vertical stretch by a (compression if 0 < a < 1)

Key facts

Inside changes move the graph sideways, in the opposite direction to the sign: f(x + 4) shifts left 4.
Outside changes move it up or down, in the direction of the sign: f(x) − 1 shifts down 1.
Shifts are translations: the graph keeps its shape and size. Reflections flip it. Stretches change its steepness.
The vertex form of a quadratic, a(x − h)² + k, is the parent x² transformed.
Track key points: if (2, 5) is on f, then (2 + h, 5 + k) is on f(x − h) + k.
Do stretches and reflections before shifts when you build a graph step by step.

General transformation

g(x) = a · f(x − h) + k

h: horizontal shift (right if positive)
k: vertical shift (up if positive)
a: vertical stretch factor; a negative a also reflects across the x-axis

Changes outside f act on y and do what they say. Changes inside act on x and do the opposite: f(x − 3) moves right, f(2x) squeezes.

Try it: shift a graph's key points

Interactive

The triangle’s corners are three key points of the graph y = 2 − |x|: (−2, 0), (0, 2) and (2, 0). Translate it to see f(x − h) + k: every point moves right h and up k, so the peak (0, 2) lands at (h, 2 + k).

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Transformation of Functions examples, step by step

Example 1: A horizontal shift

Warm-up

Describe how the graph of g(x) = (x + 3)² relates to f(x) = x². Where is the vertex of g?

g(x) = f(x + 3) = f(x − (−3)), so h = −3.
An inside change moves the graph opposite to its sign: left 3.
The vertex (0, 0) moves to (−3, 0).

Answer: The graph shifts left 3; the vertex of g is (-3, 0).

Example 2: Several transformations

Core

List the transformations that take f(x) = |x| to g(x) = −2|x − 1| + 4, and find the corner point of g.

x − 1 inside: shift right 1.
Factor 2: vertical stretch by 2 (twice as steep).
Negative sign: reflect across the x-axis (the V opens down).
+4 outside: shift up 4. The corner (0, 0) moves to (1, 4).

Answer: Right 1, stretch by 2, reflect over the x-axis, up 4; the corner of g is (1, 4).

Example 3: Move a point

Test-style

The point (2, 5) is on the graph of y = f(x). Which point must be on the graph of y = f(x + 4) − 1?

f(x + 4) shifts left 4: x becomes 2 − 4 = −2.
−1 outside shifts down 1: y becomes 5 − 1 = 4.
Check: at x = −2, f(−2 + 4) − 1 = f(2) − 1 = 5 − 1 = 4.

Answer: The point (-2, 4) is on the new graph.

Common mistakes

✗ Mistakef(x − 3) shifts the graph left 3.

✓ Insteadf(x − 3) shifts it right 3.

To get the old output at the old point x = 0, you now need x − 3 = 0, so x = 3. Everything happens 3 units later, which is to the right.

✗ Mistakef(x) + 2 and f(x + 2) are the same move.

✓ Insteadf(x) + 2 moves up 2; f(x + 2) moves left 2.

Outside the function changes the output (vertical); inside changes the input (horizontal).

✗ Mistake−f(x) reflects across the y-axis.

✓ Instead−f(x) reflects across the x-axis; f(−x) reflects across the y-axis.

Negating the output flips every y-value, turning the graph upside down. Negating the input swaps left and right.

Not to be confused with

Transformation of Functionsvs.Transformation (geometry)

A geometric transformation moves a figure on the plane, such as a triangle. A function transformation changes an equation so its whole graph moves; the moves have the same names.

Transformation of Functionsvs.Composition of functions

A transformation is a special composition with simple linear functions, like f(x − 3). Composition in general plugs any function into another, such as f(g(x)) with g(x) = x².

Quick check: 3 questions

Question 1 of 3

How does the graph of y = f(x) − 6 compare with y = f(x)?

Show answer and explanation

Answer: A. The −6 is outside the function, so it lowers every output by 6: down 6. Left or right would need a change inside, as in f(x − 6).

Question 2 of 3

Which equation shifts y = √x left 2 and up 5?

Show answer and explanation

Answer: C. Left 2 means x + 2 inside; up 5 means +5 outside. √x − 2 + 5 moves right, √x + 5 + 2 swaps the numbers, and −5 moves down.

Question 3 of 3

The graph of y = f(x) passes through (3, −2). Which point is on y = −f(x)?

Show answer and explanation

Answer: D. Negating the output keeps x = 3 and changes y = −2 to 2. (−3, −2) is the y-axis reflection f(−x); (−3, 2) does both; (2, 3) swaps the coordinates.

Transformation of Functions: frequently asked questions

Why does f(x – h) move the graph right instead of left?

Because the new function has to reach each old input later. In g(x) = f(x − 3), the output that f made at x = 0 now happens when x − 3 = 0, at x = 3. Every point arrives 3 units further right. That is why changes inside the parentheses work opposite to their sign.

What is a parent function?

A parent function is the simplest member of a family of graphs, such as y = x², y = |x|, y = √x or y = 2^x. Every other function in the family is a transformation of it. For example, y = 3(x + 1)² − 4 is the parent x² shifted left 1, stretched by 3 and shifted down 4.

In what order should you apply transformations?

For y = a · f(x − h) + k, a safe order is: horizontal shift, then stretch or reflection, then vertical shift. Apply them to a few key points of the parent graph. For y = −2(x − 1)² + 3, the vertex (0, 0) goes right to (1, 0), stays put under the stretch and flip, then goes up to (1, 3).

Practice and tools

Function GrapherGraph a function and its transformations together

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.B.3

Cite this entry

Effortless Math Team. “Transformation of Functions.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/transformation-of-functions/

Flashcard

TermTransformation of Functionsa · f(x − h) + k

DefinitionChanging a function’s rule to shift, flip or stretch its graph. f(x − 3) + 2 moves the graph right 3 and up 2.

Study the full deck

Browse all terms in the Math Dictionary · More functions terms · Random term

TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryTranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryVertex FormVertex form is a way to write a quadratic function as y = a(x − h)² + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)² + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Parent FunctionA parent function is the simplest function of a family, before any shifts, stretches or reflections. For example, y = x² is the parent of every quadratic, and y = |x| is the parent of every absolute value function. Every other member of the family is a transformation of its parent’s graph.Functions · Algebra 1, Algebra 2, PrecalculusFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8Composition of FunctionsA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x², then f(g(2)) = f(3) = 9. Order matters.Functions · Algebra 2, PrecalculusDilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, Geometry

Transformation of Functions

What is a transformation of a function?

Transformation of Functions in a picture

Formal definition and notation

Key facts

General transformation

Try it: shift a graph's key points

Transformation of Functions examples, step by step

Example 1: A horizontal shift

Example 2: Several transformations

Example 3: Move a point

Common mistakes

Not to be confused with

Quick check: 3 questions

Transformation of Functions: frequently asked questions

Why does f(x – h) move the graph right instead of left?

What is a parent function?

In what order should you apply transformations?

Related terms

Practice and tools

Sources

Flashcard

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