Graphing Quadratic Functions — Parabola Guide

Quadratic functions produce parabolas — the U-shaped curves that model everything from projectile motion to the shape of satellite dishes. This guide covers graphing quadratics, finding key features, and understanding transformations using the free online graphing calculator.

Standard Form: y = ax² + bx + c

The three coefficients control different aspects of the parabola:

• a — controls the width and direction. If a > 0, parabola opens upward. If a < 0, it opens downward. Larger |a| = narrower curve.
• b — shifts the vertex left or right
• c — the y-intercept (where the curve crosses the y-axis)

Try these in the calculator:

• y = x^2 — basic upward parabola, vertex at (0, 0)
• y = -x^2 — opens downward
• y = 2*x^2 — narrower
• y = 0.5*x^2 — wider

Vertex Form: y = a(x − h)² + k

Vertex form makes it easy to read off the vertex directly. The vertex is at (h, k).

• y = (x - 2)^2 + 3 — vertex at (2, 3), opens upward
• y = -(x + 1)^2 - 2 — vertex at (−1, −2), opens downward

Finding the Vertex from Standard Form

When given y = ax² + bx + c, the x-coordinate of the vertex is:

x = −b / (2a)

For y = x^2 - 4x + 3: x = −(−4) / (2×1) = 2. Then y = 4 − 8 + 3 = −1. Vertex: (2, −1).

Finding Roots (x-Intercepts)

The roots are where the parabola crosses the x-axis (y = 0). Use the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

The discriminant (b² − 4ac) tells you how many real roots exist:

• > 0 — two real roots (parabola crosses x-axis twice)
• = 0 — one real root (parabola just touches x-axis)
• < 0 — no real roots (parabola doesn't cross x-axis)

The graphing calculator marks roots automatically — enter y = x^2 - 4x + 3 and you'll see root markers at x = 1 and x = 3.

Transformations of Parabolas

Use sliders to explore transformations dynamically. Enter:

y = a*(x - h)^2 + k

Sliders for a, h, and k appear automatically. Adjust each to see:

• a slider — narrows/widens the curve, flips it upside-down
• h slider — moves the vertex left and right (horizontal shift)
• k slider — moves the vertex up and down (vertical shift)

Axis of Symmetry

Every parabola is symmetric about a vertical line through its vertex, called the axis of symmetry: x = h (or x = −b/(2a) in standard form).

Plot the axis of symmetry alongside the parabola:

• Equation 1: y = x^2 - 6x + 5
• Equation 2: x = 3 (vertical line at axis of symmetry)

Real-World Application: Projectile Motion

Projectile motion follows a quadratic path. The height h of an object thrown upward with initial velocity v₀ from height h₀ is:

y = -4.9*x^2 + 20*x + 1.5

Where x is time (seconds) and y is height (metres). Enter this in the calculator to visualise when the object reaches maximum height and when it hits the ground.