Quadratic Functions — Everything You Need to Know

Quadratic functions appear everywhere — from the arc of a thrown ball to the design of suspension bridges. Master this topic and many others become far easier.

MathPal Team

February 22, 2026

Quadratic functions are one of the most important foundations in algebra. Once you truly understand them, a huge range of other problem types become much more approachable.

General form

A quadratic function takes the form:

f(x) = ax² + bx + c (a ≠ 0)

Its graph is a parabola — opening upward when a > 0, downward when a < 0.

3 Important Representations

1. Standard form

f(x) = ax² + bx + c

Easy to identify coefficients and find the y-intercept at (0, c).

2. Vertex form

f(x) = a(x−h)² + k

The vertex of the parabola is at V(h, k). Makes it easy to spot the maximum or minimum value.

3. Factored form

f(x) = a(x−x₁)(x−x₂)

Used when the two roots x₁, x₂ are known. Immediately reveals the x-intercepts.

Formulas You Must Know

x-coordinate of the vertex: xₘ = −b / (2a)

y-coordinate of the vertex: yₘ = f(xₘ) = c − (b² / (4a))

Discriminant: Δ = b² − 4ac

  • Δ > 0: two distinct real roots x₁,₂ = −b ± √Δ / (2a)
  • Δ = 0: one repeated root x = −b / (2a)
  • Δ < 0: no real roots

Quick Graphing Technique

To sketch a parabola accurately, just identify 5 points:

  1. Vertex V(h, k)
  2. y-intercept: (0, c)
  3. The point symmetric to (0, c) across the axis of symmetry
  4. x-intercepts (if any): x₁, x₂

The axis of symmetry is the vertical line x = h = −b / (2a).

Real-world Applications

Quadratic functions describe many natural phenomena:

  • Physics: projectile motion, free fall under gravity
  • Engineering: suspension bridge design, parabolic antennas
  • Economics: revenue optimization, break-even analysis

Example: A ball is thrown upward with initial velocity v₀ = 20 m/s from height h₀ = 1.5 m. Its height over time:

h(t) = −5t² + 20t + 1.5

The vertex gives the maximum height and the time at which it occurs.

Practice with MathPal

Take a photo of your quadratic function problem and send it to MathPal — the AI will work through it step by step, finding the vertex, sketching the graph, solving the equation, with clear explanations at every stage.