Horizontal Shifts

Pre-Calculus · Axiom Academy

Replacing x with (x - h) slides a graph sideways — and the direction is the opposite of the sign you see. Start with the parabola f(x) = x^2 , its vertex parked at the origin. Now build g(x) = (x - 3)^2 and watch what happens: the whole curve glides 3 units to the right , carrying its vertex from (0,0) to (3, 0) . Same shape, new address. Shifted right 3 — vertex at (3,0) 2. Why the Sign Reads Backwards The graph doesn't "see" the formula — it only cares about what input makes the inside zero (and what input reproduces each old output). For g(x) = (x-3)^2 , the inside hits 0 when x = 3 . So the height f produced at input 0 now requires input 3 : every point has to travel +3 to the right to "catch up" to the value it used to have. To get the output f had at x = a , the new graph f(x - h) needs x = a + h . Bigger input required the curve sits farther right. The vertex of x^2 sits at x = 0 . (3 - 3)^2 = 0 , so the vertex moved to x = 3 . A second checkpoint on the original. (4 - 3)^2 = 1 — again a jump of +3 . 3. Left or Right, Set by the Sign Write every horizontal shift in one canonical form, g(x) = f(x - h) . Then the single number h controls everything: positive h shifts right, negative h shifts left. Slide h from -3 up to +3 and the parabola walks from the left side of the plane to the right, its vertex tracking x = h exactly.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.