Quadratic Equations & Parabolas

Explore the geometry of quadratic functions. Connect algebraic symbols directly to geometric curves, find zeroes, and calculate complex roots.

Interactive Sandbox

Parabola Graph Sandbox

Toggle between Standard and Vertex forms. Drag sliders to translate, reflect, stretch, and compress the parabola. Hover your cursor over the graph to examine the geometric properties of a parabola!

Equation Display

y = 1.00x² + 0.00x + 0.00
Coefficient a (Shape & Direction) 1.00
Coefficient b (Horizontal Tilt) 0.00
Coefficient c (Y-Intercept) 0.00
A parabola is defined as all points equidistant from the Focus and the Directrix. Try turning on the Focus & Directrix checkbox and hovering your cursor over the graph to see this geometry in real time.
Mathematical Solver

Zeros & The Discriminant

Solving the equation y = 0 means finding the x-intercepts (roots). The discriminant (D = b² - 4ac) dictates how many intersections exist. Observe the formula evaluated in real-time below.

Solving: 1.00x² + 0.00x + 0.00 = 0


1. Identify coefficients:
a = 1.00, b = 0.00, c = 0.00
2. Calculate the Discriminant (D):
D = b² - 4ac
D = (0.00)² - 4(× 1.00 × 0.00)
D = 0.00
3. Evaluate Roots using Quadratic Formula:
x = [-b ± √D] / [2a]
x = [- (0.00) ± √(0.00)] / [2 × 1.00]
x = 0.00

Root Characterization


One Real Root (D = 0)

Vertex Coordinates

(0.00, 0.00)

Calculated Zeros

x₁ = 0.00

When the discriminant D < 0, the quadratic equation has no real roots because it never crosses the x-axis. Its zeros are complex conjugate pairs containing the imaginary unit i (where i² = -1).
Interactive Physics Game

Trajectory Projectile Launch

Parabolas describe the paths of objects flying under the influence of gravity. Select a challenge level, adjust the launch angle and initial speed to shape the trajectory equation, and hit launch to hit the target basket!

Success! Target Cleared.

Trajectory Function

y = -0.10x² + 0.50x + 0.00
Launch Angle (θ) 45°
Launch Speed (v₀) 12.0
Gravity is pulling the projectile down, giving the vertical position a negative quadratic term a < 0. Adjust both the launch velocity and launch angle to target the basket.
Self Evaluation

Conceptual Quiz

Test your understanding of vertices, coefficients, translations, and discriminants. Solve the questions below to test your mastery of quadratic equations.

Question 1 of 5 Score: 0/0

For the parabola y = 2(x - 3)² + 5, what are the coordinates of the vertex?

Correct Answer!

Indeed, in vertex form y = a(x - h)² + k, the vertex coordinates are (h, k). Since the equation is y = 2(x - 3)² + 5, h = 3 and k = 5, yielding vertex (3, 5).

Conceptual Focus Areas


  • Vertex Identification
  • Discriminant Classification
  • Horizontal & Vertical Shifts
  • Complex Conjugate Roots