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Quadratic Functions & Parabolas Lab

Explore how the coefficients of a quadratic function shape its parabola. Switch between standard, vertex, and intercept forms to see how each reveals different properties. Find the vertex, roots, discriminant, and axis of symmetry, all updated live as you adjust parameters.

Guided Experiment: Discriminant and Root Classification

How does the discriminant b² − 4ac determine the number and type of roots a quadratic equation has? What happens visually to the parabola in each case?

Write your hypothesis in the Lab Report panel, then click Next.

Graph

-2-1012345670510152025(2, 0)(3, 0)Vertex (2.5, -0.25)(0, 6)x = 2.5VertexRootsy-intercept

Controls

Standard (ax² + bx + c)

Leading coefficient (a)
1
-55
Standard form coefficients

Equation Forms

Standard form
y=x2−5x+6y = x^{2} - 5x + 6
Vertex form
y=(x−2.5)2−0.25y = (x - 2.5)^{2} - 0.25
Intercept form
y=(x−2)(x−3)y = (x - 2)(x - 3)

Properties

Vertex
(2.5, −0.25)(2.5,\, -0.25)
Axis of symmetry
x=2.5x = 2.5
y-intercept
(0, 6)(0,\, 6)
Direction
Opens upward ↑
Minimum value
y=−0.25y = -0.25
Discriminant
Δ=1  (>0)\Delta = 1 \;(> 0)

Roots (x-intercepts)

Two distinct real roots
2,32, \quad 3
Via quadratic formula
x=−b±b2−4ac2a=5±12x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} = \frac{5 \pm \sqrt{1}}{2}

Completing the Square

1Start with standard form
y=x2−5x+6y = x^{2} - 5x + 6
2Take half of the x coefficient (-5), square it to get 6.25
(−52)2=6.25\left(\frac{-5}{2}\right)^{2} = 6.25
3Add and subtract this value to complete the square
y=(x−2.5)2−0.25y = (x - 2.5)^{2} - 0.25
4Vertex form with vertex at (2.5, -0.25)
y=(x−2.5)2−0.25y = (x - 2.5)^{2} - 0.25

Data Table

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Reference Guide

Three Forms of a Quadratic

Every quadratic function can be written in multiple equivalent forms, each highlighting different properties.

Standard: y=ax2+bx+c\text{Standard: } y = ax^{2} + bx + c
Vertex: y=a(x−h)2+k\text{Vertex: } y = a(x - h)^{2} + k
Intercept: y=a(x−p)(x−q)\text{Intercept: } y = a(x - p)(x - q)

The Quadratic Formula

The roots of any quadratic are given by the quadratic formula, which depends on the discriminant.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

When the discriminant is positive there are two real roots, when zero there is one repeated root, and when negative there are no real roots.

Completing the Square

Completing the square converts standard form to vertex form, revealing the vertex of the parabola directly.

x2−4x+3=(x−2)2−1x^{2} - 4x + 3 = (x - 2)^{2} - 1

The vertex is at (h, k), where h and k come from the completed square form.

Vertex and Axis of Symmetry

The vertex is the highest or lowest point on the parabola, located at the axis of symmetry.

h=−b2a,k=f(h)=ah2+bh+ch = \frac{-b}{2a}, \quad k = f(h) = a h^{2} + bh + c

If a is positive the vertex is a minimum. If a is negative it is a maximum. The axis of symmetry is the vertical line x = h.