Quadratic Equations and Parabolas: SAT Practice Questions & Study Guide

Quadratic equations and their associated parabolas are arguably the most heavily tested topic in the Advanced Math section. A quadratic function f(x) = ax^2 + bx + c produces a parabola that opens upward when a > 0 and downward when a < 0. The key features are the vertex (the maximum or minimum point), the axis of symmetry, the y-intercept (c), and the x-intercepts (zeros, found by solving ax^2 + bx + c = 0).

The vertex form f(x) = a(x - h)^2 + k directly reveals the vertex (h, k) and the axis of symmetry x = h. Converting from standard to vertex form requires completing the square. The vertex x-coordinate can also be found quickly using h = -b/(2a) without completing the square; then substitute to find k = f(h). The Digital SAT frequently gives a quadratic in a real-world context (projectile height, area optimization) and asks for the maximum or minimum value—that's always the y-coordinate of the vertex.

For finding zeros, the most important concept is the relationship between the discriminant (b^2 - 4ac) and the number of x-intercepts: positive discriminant means two real zeros (parabola crosses the x-axis twice), zero discriminant means the vertex touches the x-axis exactly once (one repeated zero, the parabola is tangent to the axis), and negative discriminant means no real zeros (the parabola doesn't touch the x-axis). The SAT tests this graphically (asking which graph has a given discriminant property) and algebraically (asking for the value of a parameter that gives exactly one solution).

Vieta's formulas provide a shortcut for sum and product of roots: for ax^2 + bx + c = 0, the sum of roots is -b/a and the product of roots is c/a. If the SAT gives you two roots and asks for a coefficient, these relations let you write the answer immediately without using the quadratic formula.