Conic Sections Basics (Parabolas, Ellipses)

ACT Math · Axiom Academy

Parabolas and ellipses, ACT-style — the two equations you need and how to read a vertex or a center straight off the numbers. 1. Parabolas — the Vertex Is Sitting Right in the Equation A parabola is the graph of a quadratic — you've already met it in algebra. Written in vertex form , the vertex's coordinates are two of the three numbers in the equation, no completing the square required. Vertex: (h, k) · opens up if , down if · narrower if , wider if Find the vertex of y = -2(x-3)^2 + 5 . Match it to the pattern: a=-2 , h=3 , k=5 . The vertex is just (h,k) — read straight off: (3, 5) . Since , the parabola opens down ; since , it's narrower than the plain y=x^2 parabola. Occasionally the ACT flips the roles of x and y : x = a(y-k)^2 + h opens left (if ) or right (if ) instead of up/down — same vertex (h,k) , same a -sign logic, just sideways. 2. Ellipses — a Circle Stretched Along One Axis An ellipse is what you get when a circle's radius grows at a different rate along each axis. In standard form, the center and the two semi-axis lengths are all sitting directly in the equation. Center: (h, k) · a = horizontal semi-axis · b = vertical semi-axis Identify the center and shape of . Center: (1, -2) . Under the (x-1)^2 term, a^2=16 , so a=4 (horizontal semi-axis). Under the (y+2)^2 term, b^2=9 , so b=3 (vertical semi-axis). Since ( ), the ellipse is wider horizontally than it is tall.

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