Quadratic Functions and Parabolas

A quadratic function has the form f(x) = ax² + bx + c. Its graph is a parabola — a symmetric U-shaped curve. Understanding the shape and position of a parabola is crucial for SAT® word problems and graph interpretation.

The vertex form f(x) = a(x - h)² + k is the most informative format. The vertex is the point (h, k) — the maximum or minimum of the function. The axis of symmetry is the vertical line x = h.

Interpreting a, h, and k in context:

• a > 0: parabola opens upward (minimum at vertex)
• a < 0: parabola opens downward (maximum at vertex)
• |a| > 1: parabola is narrower than y = x²
• |a| < 1: parabola is wider than y = x²
• h shifts the vertex left/right; k shifts it up/down

The zeros (x-intercepts) of a parabola are where f(x) = 0. You can find them by factoring, using the quadratic formula, or reading them from a graph. A parabola may have 0, 1, or 2 real zeros depending on the discriminant.

To convert from standard form to vertex form, complete the square or use h = -b/(2a) and k = f(h).

⚠️ Common mistake: confusing the sign of h. In f(x) = (x - 3)², the vertex is at x = +3, not x = -3.

💡 Tip: On the SAT®, if a question gives you a graph and asks for the equation, identify the vertex (h, k) and one other point to solve for a.