1.2.2—Related graphs

Graph transformations change coordinates systematically: y=f(x)+a shifts vertically, y=f(x−a) shifts right, y=−f(x) reflects in the x-axis, and y=f(−x) reflects in the y-axis.

For y=af(bx+c)+d, apply the inside transformation before the outside one: horizontal scale/reflection and shift act on x, while a and d act on y. Track a known point to check the direction.

A point (2,3) on y=f(x) becomes (5,3) on y=f(x−3), because the input required to produce the same output is three larger.

The sign inside f changes the horizontal direction opposite to the visible sign outside; y=f(x−3) shifts right, not left.