AHL 2.8 (HL)—Graph transformations

For y=f(x), adding outside the function changes outputs: y=f(x)+b shifts vertically. Changing the input changes where an existing feature occurs: y=f(x−a) shifts right by a.

A factor outside, y=af(x), stretches or reflects vertically; a factor inside, y=f(bx), changes horizontal scale by 1/|b| and may reflect when b<0. Composite transformations must be applied in the stated order.

From y=x², the graph y=2(x−3)²+1 has vertex (3,1), opens upward and is twice as steep vertically. Read the vertex after the horizontal shift, then the vertical changes.

Inside and outside signs behave differently: f(x−3) moves right, while f(x)+3 moves up. Do not reverse a horizontal scale factor.