A faint grey dashed curve always shows the original, untransformed function so you can see exactly what each parameter changed. The live equation and a plain-English description of the transformation (e.g. "flipped vertically, shifted right 2, shifted up 3") update as you drag. Try the guided tour for the four transformation rules in order, or the quiz to practise evaluating a transformed function at a given x.
- What does the a do in y = a·f(x)?
- a stretches the graph vertically by a factor of a: every y-value is multiplied by a. If a is negative, the curve also flips upside down (reflects in the x-axis). a = 2 doubles the height of every point; a = 0.5 halves it.
- What does a negative b do to a graph?
- b multiplies x before the function is applied. A negative b reflects the graph left–right across the y-axis, swapping its left and right halves. Combined with a fractional size, e.g. b = -0.5, it both flips and stretches the graph horizontally.
- How do you shift a graph left or right?
- Use h in f(x − h): a positive h shifts the curve right by h units, and a negative h shifts it left. This often confuses students because of the minus sign — f(x − 3) moves right, not left, by 3.
- What is the difference between a horizontal and a vertical stretch?
- A vertical stretch (changing a) multiplies every y-value, making the graph taller or shorter while the x-axis positions stay the same. A horizontal stretch (changing b) multiplies the x-values needed to reach each y-value, making the graph wider or narrower while the y-axis heights stay the same.
- In what order do multiple transformations apply?
- Work from the inside out: first apply b (horizontal stretch/reflect) and h (horizontal shift) to x inside the function, then apply a (vertical stretch/reflect) and k (vertical shift) to the result. Doing them in the wrong order gives a different, incorrect graph.
- Why does b seem to stretch the graph the "wrong way"?
- Because b multiplies x before f is applied, a b between 0 and 1 (e.g. 0.5) means you need a bigger x to get the same input to f, so the graph stretches outward. A b greater than 1 compresses it inward — the opposite of how a behaves on the y-axis.