Vertical Shifts

Pre-Calculus · Axiom Academy

Add a constant to a function and its whole graph slides straight up or down — same shape, new height. 1. Every Point Moves the Same Amount Start with f(x) = x^2 (gray) and add 2 to get g(x) = x^2 + 2 (blue). Pick any input x — the new height is just the old height plus 2 . Watch the matching arrows: at every x the lift is identical, so the whole curve rises rigidly without bending. Shifted up by 2 — every height gains 2 2. The Sign of k Sets the Direction The constant k is a dial. Make it positive and the graph slides up ; make it negative and it slides down . In the animation the dashed gray parabola marks the starting position f(x) = x^2 , and the blue parabola glides up to k = +3 , back down through k = -2 , and home to k = 0 — never changing shape. Adding a positive constant raises every output, so the graph moves UP by k units. Subtracting (a negative k ) lowers every output, so the graph moves DOWN by |k| units. Add nothing and the graph stays exactly where it started: g(x) = f(x) . Width, openness, and turning point never change — a shift only relocates the curve. The vertex of y = x^2 sits at (0, 0) . Under g(x) = x^2 + 3 it climbs to (0, 3) ; under g(x) = x^2 - 3 it drops to (0, -3) . The vertex just tracks k straight up or down the y -axis. 3. k Is Exactly the y -Intercept Shift

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