Functions
AI Assistant

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)."

Graph

FAQ

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.

Related Topics

Quadratic EquationsAbsolute Value FunctionsWhat is a Function?