Function Transformations

Shift, stretch, and reflect any function with sliders

Every function can be transformed using the formula y = a · f(x − h) + k. The parameter k shifts the graph up or down, h shifts it left or right, and a stretches or compresses it vertically. When a is negative, the graph flips upside down.

Use the sliders below to experiment: start with the k slider to shift vertically, then try h for horizontal shifts, and finally a for stretching and reflection. Watch how the shape changes in real time.

Ask the AI "What does a = −2 do?" or "Move the vertex to (3, 4)."

Related Topics

Quadratic EquationsAbsolute Value FunctionsWhat is a Function?

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What does the k parameter do?
The parameter k in y = f(x) + k shifts the entire graph vertically. Positive k moves it up, negative k moves it down. The shape stays exactly the same — it just slides up or down.
Why does h shift in the opposite direction?
In y = f(x − h), the graph shifts RIGHT by h units (not left). This is because x needs to be h units larger to produce the same output. Think of it as "delaying" the function by h.
What is a vertical stretch?
When |a| > 1 in y = a·f(x), the graph gets taller (stretched vertically). When 0 < |a| < 1, it gets shorter (compressed). The x-intercepts stay the same, but every other point moves farther from or closer to the x-axis.
What happens when a is negative?
A negative value of a in y = a·f(x) reflects the graph across the x-axis — it flips upside down. A parabola that opened upward now opens downward. Combined with |a| ≠ 1, you get both reflection and stretching.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.