Function Transformation Calculator

Analyze transformations of parent functions using y = a·f(x − h) + k, including vertical stretch, reflection, and horizontal or vertical shifts.

FunctionsTransformationsPrecalculus

Calculate

Parent function f(x)

Vertical factor a

Horizontal shift h

Vertical shift k

Please enter valid values.

What this calculator teaches

Transformations let students build new graphs from familiar parent functions. The form y = a·f(x − h) + k tells you almost everything: a changes height and reflection, h shifts left/right, and k shifts up/down.

This page is especially helpful for Algebra 2, Precalculus, and graphing lessons because it explains the meaning before the calculation.

Formula & Symbols

Concept	Formula	Meaning
General transformation	`y=a f(x-h)+k`	a controls vertical stretch/reflection, h shifts horizontally, k shifts vertically.
Horizontal shift	`x-h`	Positive h moves the graph right; negative h moves it left.
Vertical shift	`+k`	Positive k moves the graph up; negative k moves it down.

Step-by-Step Examples

Example 1

Transforming a parabola

Start with f(x)=x².
Use a=2, h=3, k=-1.
The graph moves right 3, down 1, and becomes vertically stretched by 2.

✓ y = 2(x - 3)² - 1.

Example 2

Reflecting an absolute-value graph

Start with f(x)=|x|.
Use a=-1, h=0, k=4.
The negative a reflects the V-shape across the x-axis, then k moves it up 4.

✓ y = -|x| + 4.

Where students use this

🎓

High school and college

Use the tool to check homework and understand each step.

📊

Data and modeling

Connect formulas to tables, graphs, and real values.

🧪

Science and engineering

Use math results inside physics, chemistry, and engineering problems.

💻

Computer science

Apply the same logic to algorithms, systems, and numerical work.

Common Mistakes to Avoid

⚠️

Wrong direction for h

In f(x - h), positive h shifts right, not left.

⚠️

Ignoring negative a

A negative a reflects the graph over the x-axis.

⚠️

Mixing stretch and shift order

Describe transformations carefully: parent function first, then horizontal shift, vertical stretch/reflection, and vertical shift.

Frequently Asked Questions

What does a do in y = a f(x-h)+k? ▾

It stretches or compresses the graph vertically. If a is negative, it also reflects the graph across the x-axis.

Why does x - 3 move right? ▾

Because the transformed function reaches the old input value when x is 3 units larger.

What is the vertex after transformation? ▾

For x² or |x| parent functions, the vertex moves to (h,k).

Can this apply to square-root functions? ▾

Yes. The starting point of √x moves from (0,0) to (h,k).

Related Math Calculators

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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