Reflections and Combined Transformations

Calculus Readiness · Axiom Academy

LESSON Reflections and Stretches A single minus sign flips a graph across an axis; a single scale factor stretches or squeezes it. Four moves, shown as the motion that defines them. 1. Reflection Over the x-axis: -f(x) A reflection is a mirror image — fold the paper along a line, trace, and the new curve is what you get. To flip over the x-axis , negate the output : every height b becomes -b , so the point (a, b) drops to (a, -b) . The x-coordinates never move. Every point reflects straight down through the axis 2. Reflection Over the y-axis: f(-x) To flip over the y-axis , negate the input instead. Replacing x with -x sends the point (a, b) across to (-a, b) : the heights are unchanged, but left and right trade places. For a one-sided graph like , the whole curve swings to the other half-plane. Same height, mirrored left ↔ right An even function satisfies f(-x) = f(x) : reflecting over the y-axis gives back the same graph. An odd function satisfies f(-x) = -f(x) : reflecting through the origin gives back the same graph. To test, compute f(-x) and simplify — equals f(x) means even, equals -f(x) means odd, otherwise neither. Multiplying the whole output by a constant a rescales the graph vertically . Every point (a, b) becomes (a, k b) : heights are multiplied by k , while the x-coordinate stays put. If k > 1 the curve stretches taller; if 0 < k < 1 it compresses flatter. Output scaled by k — taller when k>1 , flatter when k<1

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