ƒ(x) Function Transformation Calculator

Analyze parent-function transformations with equations, graph behavior, mapped points, domain, range, intercept estimates, and detailed step-by-step reasoning.

Inputs

Model: y = a f(b(x − h)) + k. Use fractions or pi expressions like 1/2 or pi/4.

Steps
Result
y = 2(x − 3)² − 4 | Domain: all real numbers | Range: [-4, ∞)
Transformation model
\(y=a\,f(b(x-h))+k\)
Graph comparison

The faint curve is the parent function. The darker curve is the transformed function. Points mark key landmarks or table samples.

Calculation Steps
7 shown
Setup: Parent FunctionUse the Transformation ModelHorizontal TransformationVertical TransformationDomain and RangeKey FeaturesIntercept Check
1Setup: Parent Function\(y=f(x)\)
Before
Parent: Quadratic | f(x) = x²
After
Apply transformation model
Start with the parent function Quadratic. Its base rule is f(x) = x².
Formula\(y=f(x)\)
Before
a=2, b=1, h=3, k=-4
After
y = 2(x − 3)² − 4
The outside coefficient a controls vertical stretch and reflection. The inside coefficient b controls horizontal compression and reflection. The values h and k shift the graph.
Formula\(y=a\,f(b(x-h))+k\)
Before
Parent input x
After
New input: x − 3
The graph shifts by h = 3. Then the input is scaled by b = 1. A negative b reflects the graph across a vertical line through the shift.
Formula\(x\mapsto b(x-h)\)
Before
Parent output f(x)
After
Output: 2f(input) − 4
The graph is vertically scaled by |a| = 2. No vertical reflection occurs because a is positive. The final vertical shift is k = -4.
Formula\(y\mapsto a y+k\)
Before
Parent restrictions are transformed
After
Domain: all real numbers; Range: [-4, ∞)
The domain changes when the parent function restricts its input. The range changes after vertical stretch, reflection, and vertical shift.
Formula\(\text{Domain from input restrictions; range from output changes}\)
Before
Parent graph landmarks
After
turning point at (3, -4); opens upward; axis of symmetry x = 3
Main graph features: turning point at (3, -4); opens upward; axis of symmetry x = 3.
Formula\(\text{features move with }(h,k)\)
Before
Set x = 0 and y = 0
After
y-intercept: (0, 14); x-intercepts: 1.585786, 4.414214
The y-intercept comes from substituting x = 0. The x-intercepts are solved numerically inside the default graph window.
Formula\(x\text{-intercepts: }y=0,\quad y\text{-intercept: }x=0\)
Worked Examples
General2(x−3)²−4Mapu=2 under quadratic transformVertex−(x+2)²+5Absolute3|x−4|−2Radical−2√(x−1)+6Exponential3·2^(x−1)−4Log2log₁₀(x−3)+1Trig3sin(2(x−π/4))−1TableAbsolute value tableSequenceReflect and stretch cubic
Transformation Tools
General
Full parent-function model
Point Map
Move parent coordinates to new graph
Quadratic
Vertex, axis, roots, range
Absolute
Corner, arms, intercepts
Radical
Endpoint, domain, range
Exponential
Asymptote and growth behavior
Logarithmic
Vertical asymptote and domain
Trig
Amplitude, period, phase, range
Table
Generate graph points
Sequence
Convert parameters to words
Key Rules
Master\(y=a f(b(x-h))+k\)
Vertical\(a stretches and reflects y\)
Horizontal\(b compresses by 1/|b|\)
Shift\(h moves horizontally\)
Lift\(k moves vertically\)
Point map\((u,v)→(h+u/b,av+k)\)
Quadratic\(vertex=(h,k)\)
Trig\(period=2π/|b|\)
Exp\(asymptote y=k\)
Log\(asymptote x=h\)
Input Tips
  • Use fractions such as 3/2.
  • Use pi or pi/4 for angle shifts.
  • b cannot be zero.
  • Exponential and logarithmic bases must be positive.
  • Base 1 is not allowed.
  • Log and square-root graphs may have restricted domains.
  • Table rows are limited to keep the layout clean.
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