Transformation of Graphs Calculator (Translation / Scaling / Reflection)
Enter f(x) and transform parameters to get the new g(x) expression and sample values, covering translation, scaling and reflection.
Input Data
Results
At a glance:Graph transformations: translation g(x)=f(x−h)+k shifts right h, up k; vertical scaling g(x)=b·f(x) multiplies y by b; horizontal scaling g(x)=f(x/a) multiplies x by a; reflection x-axis g=−f(x), y-axis g=f(−x), origin g=−f(−x). Key points move with the transform.
Formula
Translation: g(x) = f(x − h) + k (right h, up k).
Vertical scaling: g(x) = b·f(x).
Horizontal scaling: g(x) = f(x / a).
Reflect x-axis: g(x) = −f(x); y-axis: g(x) = f(−x); origin: g(x) = −f(−x).
$$g(x) = f(x-h) + k$$$$g(x) = b\,f(x),\quad g(x)=f(x/a)$$How to Use
- Choose the transform type.
- Enter f(x) and the relevant parameters (h, k, a, b).
- Enter a sample x to compare f and g values.
- The tool gives g(x) and a vertex-change note.
Effect on keypoint (x₀,y₀)
| Transform | New keypoint | Note |
|---|---|---|
| f(x−h)+k | (x₀+h, y₀+k) | right h, up k |
| b·f(x) | (x₀, b·y₀) | y scale |
| f(x/a) | (a·x₀, y₀) | x scale |
| −f(x) | (x₀, −y₀) | flip down |
| f(−x) | (−x₀, y₀) | flip left-right |
Horizontal scaling f(x/a): a>1 widens (x grows), opposite to vertical intuition.
Case Studies
Translation: f(x)=x² → f(x−2)+3
g(x)=(x−2)²+3, vertex moves (0,0) → (2,3).
x=1: f=1, g=(−1)²+3=4.
Vertical scaling: f(x)=x², b=2
g(x)=2x², narrower (y doubled).
Vertex unchanged (0,0); x=1: f=1, g=2.
Horizontal scaling: f(x)=x², a=2
g(x)=(x/2)²=x²/4, wider.
x=2: f=4, g=1.
Reflect x-axis: f(x)=x² → −x²
g(x)=−x², parabola flips down, vertex still (0,0).
Reflect y-axis: f(x)=x³ → (−x)³
g(x)=(−x)³=−x³, mirrored left-right (odd symmetry).
Reflect origin: f(x)=x+1 → −(−x+1)
g(x)=x−1, original rotated 180° about origin.
FAQ
f(x−h): left or right?
f(x−h) shifts right by h (to recover f(x₀) you need x=x₀+h). A common mistake is thinking 'minus = left'.
Why f(x/a) for horizontal scaling?
a>1 means x must be larger to return to the old y, so the graph widens — opposite to vertical b·f(x) intuition; watch carefully.
What is origin reflection?
g(x)=−f(−x) flips both left-right and up-down, equivalent to rotating the graph 180° about the origin.
How does the vertex move?
Translation moves it directly; vertical scaling scales y (vertex y × b); horizontal scaling scales x (× a); reflection negates coordinates.
Combine transforms?
Yes, e.g. translate then scale. Order matters: usually handle inner (x) then outer (y).
Use of sample values?
Comparing f and g at the same x quickly verifies the transform (e.g. translated y differs by k).
Works for periodic (sin)?
Yes. b·sin x changes amplitude; sin(x/a) changes period to 2πa.
Horizontal scale vs shift confusion?
f(x−h) is a shift, f(x/a) is a scale; together f((x−h)/a) means shift then scale — order is important.
Related Tools
References
Content review: Calculatorism Science Team. Translation/scaling/reflection transform logic and sample values verified. Results are for reference only.