Quadratic Function with Roots and Vertex Calculator

A quadratic function is a second-degree polynomial that graphs as a parabola. This calculator helps you find the roots (x-intercepts) and vertex of a quadratic function in the form f(x) = ax² + bx + c.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree 2. It has the general form:

f(x) = ax² + bx + c

Where:

a - Coefficient of x² (determines the parabola's width and direction)
b - Coefficient of x (affects the parabola's slope)
c - Constant term (y-intercept)

The graph of a quadratic function is a parabola. The parabola can open upwards (if a > 0) or downwards (if a < 0).

Quadratic Function Formula

The standard form of a quadratic function is:

f(x) = ax² + bx + c

This can also be written in vertex form:

f(x) = a(x - h)² + k

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

Finding the Roots

The roots of a quadratic function are the x-intercepts, where f(x) = 0. They can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: No real roots (complex roots)

Finding the Vertex

The vertex of a parabola is the point where the function reaches its maximum or minimum value. For a quadratic function in standard form, the vertex (h, k) can be found using:

h = -b / (2a)

k = f(h) = c - (b² / (4a))

The vertex form of the quadratic function is particularly useful for identifying the vertex directly.

Worked Example

Let's find the roots and vertex for the quadratic function f(x) = 2x² - 8x + 6.

Step 1: Identify coefficients

a = 2, b = -8, c = 6

Step 2: Find the roots using the quadratic formula

x = [8 ± √((-8)² - 4*2*6)] / (2*2)

x = [8 ± √(64 - 48)] / 4

x = [8 ± √16] / 4

x = [8 ± 4] / 4

So, the roots are:

x₁ = (8 + 4)/4 = 3
x₂ = (8 - 4)/4 = 1

Step 3: Find the vertex

h = -b / (2a) = 8 / 4 = 2

k = f(2) = 2(2)² - 8(2) + 6 = 8 - 16 + 6 = -2

The vertex is at (2, -2).

Step 4: Vertex form

f(x) = 2(x - 2)² - 2

FAQ

What is the difference between standard and vertex form?: The standard form (ax² + bx + c) is useful for calculating roots and intercepts, while the vertex form (a(x - h)² + k) makes it easy to identify the vertex and axis of symmetry.
How do I know if a quadratic function has real roots?: A quadratic function has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex numbers.
What does the coefficient 'a' represent in a quadratic function?: The coefficient 'a' determines the width and direction of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
Can a quadratic function have only one root?: Yes, a quadratic function can have exactly one real root when the discriminant is zero. This occurs when the parabola touches the x-axis at its vertex.
How do I graph a quadratic function?: To graph a quadratic function, first identify the vertex and axis of symmetry (x = -b/(2a)). Then plot the vertex and use symmetry to find additional points. The parabola will open upwards if 'a' is positive or downwards if 'a' is negative.