Reflections Across Axes

Pre-Calculus · Axiom Academy

LESSON Reflections Across Axes One negative sign flips a graph. Where you put it — outside or inside the function — decides whether it flips over the x-axis or the y-axis. 1. Negative Outside → Flip Over the X-Axis Take any graph y = f(x) and replace it with y = -f(x) . Every output gets its sign reversed, so a point that sat at height +3 now sits at -3 and a point at -2 jumps to +2 . Watch the curve below hinge straight down across the x-axis — that horizontal axis acts like a mirror, and the flipped copy is -f(x) . Negating the output reflects over the x-axis Each point keeps its x and flips its y 2. Negative Inside → Flip Over the Y-Axis Now replace y = f(x) with y = f(-x) . The sign lives inside , so it acts on the input before f ever runs: whatever used to happen at x = 3 now happens at x = -3 . The graph slides through the y-axis as if that vertical line were the mirror. Heights stay put; it's the left–right positions that swap. Acts on the output . Flips vertically, over the x-axis. . Acts on the input . Flips horizontally, over the y-axis. . A y-axis reflection pins any point already on the y-axis: the value f(0) never moves. If f is even , f(-x)=f(x) — the y-flip changes nothing. If f is odd , f(-x)=-f(x) — the two flips agree. A concrete case: f(x)=x^2 vs. f(x)=x^3

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