MathematicsFunctions and graphsClass 11Class 12

Graph transformations: interactive simulation

Most graphs in JEE are a familiar parent graph that has been shifted, stretched or flipped. Learn the four moves once and you can sketch y = −2(x − 3)² + 1 or y = 1/(x + 2) − 4 in seconds.

Useful for: JEE Main, A-level, SAT

Try this

Pick a parent graph, move the sliders, or drag the red point to slide the whole curve. The equation and a plain-English list of transformations update as you go.

  1. 1

    Parent x², set h = 2 and k = −1.

    What you should see

    The vertex moves to (2, −1): y = (x − 2)² − 1.

  2. 2

    Parent sin x, set b = 2.

    What you should see

    The graph is squeezed horizontally — the period halves from 2π to π.

  3. 3

    Parent 1/x, set h = 3 and k = 2.

    What you should see

    Both asymptotes move: x = 3 and y = 2.

  4. 4

    Set a = −1 for any parent.

    What you should see

    The graph reflects in the x-axis; b = −1 reflects it in the y-axis instead.

Formulas behind it

General transformation

y=a f(b(x−h))+ky = a\,f\big(b(x-h)\big) + k

Shifts

f(x−h): right by h,f(x)+k: up by kf(x-h):\ \text{right by } h, \quad f(x)+k:\ \text{up by } k

Stretches

a f(x): vertical×a,f(bx): horizontal×1ba\,f(x):\ \text{vertical} \times a, \quad f(bx):\ \text{horizontal} \times \tfrac{1}{b}

Common mistakes

  • Moving f(x − 3) to the left. A minus sign inside the bracket shifts the graph right.
  • Thinking f(2x) is twice as wide. It is squeezed to half the width.
  • Applying the shift before the stretch inside the bracket: f(2x − 6) = f(2(x − 3)) shifts right by 3, not 6.
Practise 4 questionsCheck your answer instantly, with a worked solution.Stuck on a graph transformations question?Snap it and get a whiteboard video solution.

FAQ

Why does f(x − h) move the graph right?

The new graph reaches the value f(0) when x − h = 0, i.e. at x = h. Every point arrives h units later, so the graph moves right.

What is the period of y = sin(3x)?

2π/3. Multiplying x by b squeezes the graph by a factor b, so the period becomes 2π/|b|.

How do I find the shift in y = (2x + 4)²?

Factor the coefficient of x: (2(x + 2))². The graph is squeezed by 2 and shifted left by 2.

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