Coordinate Geometry — Circles, Parabolas, Reflections

GRE Quantitative · Axiom Academy

LESSON Coordinate Geometry — Circles, Parabolas, Reflections Turn three geometric shapes into equations you can compute — a circle's center and radius, a parabola's opening direction, and a point's mirror image. 1. A Circle's Equation Is Its Center and Radius A circle is every point sitting exactly distance r from a fixed center (h,k) . Squaring that distance condition gives the equation — so h , k , and r aren't abstract letters, they are the circle: change one number and the drawn circle moves or grows to match, instantly. 2. The Sign of a Decides Which Way a Parabola Opens A parabola in vertex form is y=a(x-h)^2+k , with vertex (h,k) . The vertex never moves as a changes — only the steepness and direction do. Watch a sweep from positive, through zero, to negative: the curve opens upward, flattens to a line for an instant at a=0 , then opens downward — the sign IS the direction. The vertex is the LOWEST point; the curve rises on both sides. The parabola momentarily flattens — this is the crossover, not a real parabola. The vertex is the HIGHEST point; the curve falls on both sides. The bigger the magnitude of a , the more tightly the parabola hugs its axis. With vertex (3,1) fixed, a=2 gives y=2(x-3)^2+1 , which opens upward — moving away from the vertex, y only increases (e.g. ). Flip to a=-2 : same vertex, y=-2(x-3)^2+1 , now opens downward — y only decreases moving away ( ). 3. Reflecting Across y=x Swaps the Coordinates

This is the written version of the interactive lesson above. See the full GRE Quantitative course.