9th grade math - Algebra

Understanding Quadratic Functions

Quadratic functions form the foundation of many concepts in algebra, and their applications extend into physics, engineering, business, and beyond. This article introduces the standard form of a quadratic function, explains how to graph it, discusses solving techniques, and connects it with what you’ve already learned in previous lessons on exponents, polynomials, and factoring at StudyMath.org.

What Is a Quadratic Function?

A quadratic function is a function that can be written in the form:

f(x) = ax² + bx + c

Where:

  • a, b, and c are real numbers,
  • a ≠ 0,
  • x is the variable, and
  • the highest exponent is 2, making this a degree-2 polynomial.

Examples:
f(x) = x² + 3x + 2
f(x) = -2x² + 4x - 1

Quadratic functions create parabolas when graphed. The direction of the parabola depends on the sign of a:

  • If a > 0, it opens upward.
  • If a < 0, it opens downward.

Connection to Polynomials and Factoring

Quadratics are special cases of polynomials. From our previous article on Polynomials and Factoring, we saw how to factor expressions like:

x² + 5x + 6 = (x + 2)(x + 3)

Factoring is a key skill for solving quadratic equations. If you’re not confident with factoring, revisit the factoring guide.

Graphing Quadratic Functions

A quadratic function graphs as a parabola with these features:

1. Vertex

The vertex is the maximum or minimum point.
Use the formula: 

x = -b / (2a)

Then substitute into the function to find the y-coordinate.

2. Axis of Symmetry

The vertical line through the vertex: 

x = -b / (2a)

3. Y-Intercept

This is f(0) = c

4. X-Intercepts

Found by solving f(x) = 0. These are the “roots” or “zeros.”

Solving Quadratic Equations

1. Factoring

x² - 5x + 6 = (x - 2)(x - 3)

Set each factor to 0 to solve: x = 2, x = 3

2. Completing the Square

Convert to a perfect square form: 

x² + 6x + 9 = (x + 3)²

3. Quadratic Formula

For ax² + bx + c = 0, use: 

x = (-b ± √(b² - 4ac)) / 2a

Discriminant and Root Types

The discriminant is D = b² - 4ac

  • D > 0: 2 real roots
  • D = 0: 1 real root
  • D < 0: 2 complex roots

Real-World Applications

Quadratics are used to model:

  • Projectile motion – e.g., the path of a thrown object
  • Business optimization – maximize profit or minimize cost
  • Engineering designs – arches, stress analysis

This differs from what we learned in Exponential Functions, which grow or decay continuously, not symmetrically.

Practice Problems

  1. Find the vertex of f(x) = 2x² - 4x + 1
  2. Solve by factoring: x² - 5x + 6 = 0
  3. Use the quadratic formula: x² + 2x + 5 = 0
  4. Graph f(x) = -x² + 6x - 8
  5. Does f(x) = -3x² + x - 2 open up or down?

Summary

Quadratic functions are a powerful and versatile tool in algebra. They connect directly to your prior knowledge of polynomials and exponents.