Quadratic Function Given Roots and Vertex Calculator

This calculator helps you determine the quadratic function when you know its roots and vertex. Quadratic functions are fundamental in algebra and have applications in physics, engineering, and economics. By providing the roots and vertex, you can find the exact equation of the parabola that fits these points.

Introduction

A quadratic function is a second-degree polynomial that forms a parabola when graphed. The general form is:

f(x) = ax² + bx + c

Where:

a determines the parabola's width and direction (upwards if positive, downwards if negative)
b affects the parabola's slope and shift
c is the y-intercept

When you know the roots (x-intercepts) and vertex of a quadratic function, you can uniquely determine the equation. The vertex form of a quadratic function is particularly useful:

f(x) = a(x - h)² + k

Where (h, k) is the vertex of the parabola.

How to Use the Calculator

Enter the two roots of the quadratic function (x-intercepts)
Enter the vertex coordinates (h, k)
Click "Calculate" to generate the quadratic equation
Review the results including the standard form equation and vertex form equation
Use the interactive chart to visualize the parabola

Note: The calculator assumes the quadratic function has real roots. Complex roots are not supported in this version.

Formula

The quadratic function can be determined using the following steps:

Given roots x₁ and x₂, the sum and product of roots are:

Sum: x₁ + x₂ = -b/a

Product: x₁x₂ = c/a
Given vertex (h, k), the vertex form is:

f(x) = a(x - h)² + k
To find 'a', use the fact that the parabola passes through one of the roots. For example, using x₁:

0 = a(x₁ - h)² + k

a = -k / (x₁ - h)²
Once 'a' is known, convert to standard form:

f(x) = a(x² - 2hx + h²) + k = ax² - 2ahx + (ah² + k)

Worked Example

Let's find the quadratic function with roots at x = 1 and x = 3, and vertex at (2, -1).

Sum of roots: 1 + 3 = 4
Product of roots: 1 × 3 = 3
Using vertex form: f(x) = a(x - 2)² - 1
Substitute x = 1 (a root):

0 = a(1 - 2)² - 1 → 0 = a(1) - 1 → a = 1
Vertex form: f(x) = (x - 2)² - 1
Standard form: f(x) = x² - 4x + 4 - 1 = x² - 4x + 3

So the quadratic function is f(x) = x² - 4x + 3.

Interpreting Results

The calculator provides two forms of the quadratic equation:

Standard form (ax² + bx + c) - useful for calculations and graphing
Vertex form (a(x - h)² + k) - shows the vertex clearly and is useful for transformations

The interactive chart helps visualize the parabola, showing:

The vertex point
The roots (x-intercepts)
The y-intercept
The axis of symmetry (vertical line through the vertex)

Tip: The vertex form makes it easy to see if the parabola opens upwards or downwards based on the sign of 'a'.

FAQ

What if I only know one root and the vertex?: You can still determine the quadratic function, but you'll need to make an assumption about the other root or use additional information about the parabola's shape.
Can the calculator handle complex roots?: No, this calculator only works with real roots. For complex roots, you would need to use a different approach involving imaginary numbers.
What if the vertex is on the x-axis?: The calculator will still work, but the y-coordinate of the vertex (k) will be 0. This means the parabola passes through the origin (0,0).
How accurate are the calculations?: The calculator uses precise mathematical formulas and JavaScript's built-in arithmetic operations, so results should be accurate to within standard floating-point precision limits.
Can I use this calculator for cubic or higher-degree polynomials?: No, this calculator is specifically designed for quadratic functions (degree 2). For higher-degree polynomials, you would need a different calculator.