Interactive Parabola Graph

Adjust coefficients to see how the parabola changes in real-time

Coefficient \(a\): Controls width and direction

Coefficient \(b\): Controls horizontal position

Coefficient \(c\): Controls vertical position

Show vertex Show roots Show axis

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Graph Analysis

Current Equation: y = x² - 3x + 2

Vertex: (1.5, -0.25)

Axis of Symmetry: x = 1.5

y-intercept: (0, 2)

x-intercepts (Roots): (1, 0) and (2, 0)

Direction: Opens upward

Discriminant: 1 (Two real roots)

Understanding Parabolas

A parabola is the graph of a quadratic function. Every quadratic equation \(ax^2 + bx + c = 0\) corresponds to a parabola \(y = ax^2 + bx + c\).

Key Features

Vertex: The turning point (minimum or maximum)
Axis of Symmetry: Vertical line through vertex
Intercepts: Points where graph crosses axes
Direction: Upward if \(a > 0\), downward if \(a < 0\)
Width: Controlled by \(|a|\) (larger = narrower)

Root Interpretation

Two x-intercepts: Two real roots
One tangent point: Double root
No x-intercepts: Complex roots
y-intercept: Always at \(y = c\)

Standard Form vs Graph

\(ax^2 + bx + c = 0\) (Equation to solve) \(y = ax^2 + bx + c\) (Function to graph)

The solutions to the equation are the x-intercepts of the graph.

How Coefficients Affect the Parabola

\(a\) - Leading Coefficient

Controls: Width and direction

\(a > 0\): Opens upward (U-shaped)
\(a < 0\): Opens downward (∩-shaped)
\(|a|\) large: Narrow parabola
\(|a|\) small: Wide parabola

\(b\) - Linear Coefficient

Controls: Horizontal position

Affects vertex x-coordinate: \(x_v = -\frac{b}{2a}\)
Changes axis of symmetry position
Works with \(a\) to determine vertex

\(c\) - Constant Term

Controls: Vertical position

y-intercept: \((0, c)\)
Shifts graph up/down
Doesn't affect shape, only position

Try it yourself: Use the sliders above to see how each coefficient changes the graph!

Common Parabola Types

Standard Parabola

\(y = x^2\)

Vertex at origin (0, 0)
Opens upward
Axis: x = 0
Double root at x = 0

Shifted Parabola

\(y = (x - h)^2 + k\)

Vertex at (h, k)
Easy to identify features
Vertex form representation

Wide vs Narrow

y = 0.5x^2\) vs \(y = 2x^2

Same vertex, different widths
Larger |a| = narrower
Smaller |a| = wider

Real-World Applications

Projectile Motion

The path of a thrown object follows a parabolic trajectory. Vertex represents maximum height.

Optics

Parabolic mirrors and lenses focus light to a single point (the focus of the parabola).

Economics

Cost, revenue, and profit functions often have quadratic components with vertices indicating optimal values.

Engineering

Arch bridges, satellite dishes, and suspension cables use parabolic shapes for optimal strength and function.

Frequently Asked Questions

Why do some parabolas not cross the x-axis? +

When the discriminant \(b^2 - 4ac\) is negative, the parabola is entirely above or below the x-axis, meaning the quadratic equation has complex roots. The graph helps visualize why no real solutions exist.

How do I find the vertex from the graph? +

The vertex is the highest or lowest point on the parabola. For \(y = ax^2 + bx + c\), the vertex x-coordinate is \(x = -\frac{b}{2a}\). Plug this into the equation to find the y-coordinate.

What's the difference between parabola and quadratic? +

A quadratic is the algebraic expression \(ax^2 + bx + c\). A parabola is the geometric graph of a quadratic function. They're two representations of the same mathematical object.

Can a parabola open sideways? +

Yes! The equation \(x = ay^2 + by + c\) produces a sideways-opening parabola. However, this represents \(x\) as a function of \(y\), not the standard \(y\) as a function of \(x\).

Parabola Graph Calculator