The squared relationship
A quadratic function has x² as its highest power:
f(x) = ax² + bx + c (a ≠ 0)
Its graph is a parabola: a U-shaped curve.
Recognizing a quadratic from a table
For equally spaced x-values, the second differences are constant (and not zero).
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 1 | 4 | 11 | 22 | 37 |
| 1st differences | 3 | 7 | 11 | 15 | |
| 2nd differences | 4 | 4 | 4 |
Constant second differences → quadratic. (The second difference equals 2a, so a = 2 here.)
Features of a parabola
| Feature | Meaning |
|---|---|
| Direction of opening | up if a > 0, down if a < 0 |
| Vertex | the turning point |
| Axis of symmetry | the vertical line x = h through the vertex |
| Optimal value | the y-coordinate of the vertex: a minimum (opens up) or maximum (opens down) |
| y-intercept | c (when x = 0) |
| Zeros / x-intercepts | where y = 0: 0, 1 or 2 of them |
Finding the vertex
From standard form: x = −b/(2a), then substitute.
Example: f(x) = x² − 6x + 5
- x = 6/2 = 3
- f(3) = 9 − 18 + 5 = −4
- Vertex (3, −4); axis x = 3; minimum −4
Three forms
| Form | Equation | Shows |
|---|---|---|
| Standard | y = ax² + bx + c | y-intercept |
| Factored | y = a(x − r)(x − s) | zeros r and s |
| Vertex | y = a(x − h)² + k | vertex (h, k) |
Effect of a
- |a| > 1: narrower (vertical stretch)
- |a| < 1: wider (vertical compression)
- a < 0: opens down
Real-world quadratics
- Projectiles: h(t) = −4.9t² + v₀t + h₀
- Area: a rectangle with fixed perimeter
- Revenue: price × quantity, when quantity drops as price rises
- Bridges and satellite dishes with parabolic shapes
Worked examples
Example 1: Opens up or down: y = −2x² + 3? Down (a < 0); maximum 3.
Example 2: Axis of symmetry of y = 2x² + 8x − 1? x = −8/4 = −2.
Example 3: A ball’s height is h = −5t² + 20t. Max height? t = 2, h = 20 m.
Common mistakes
- Using first differences to identify a quadratic.
- Sign error in −b/(2a).
- Thinking every parabola crosses the x-axis.
Try it yourself
- Is x: 0, 1, 2, 3; y: 2, 3, 6, 11 quadratic?
- Vertex of y = x² + 4x + 1?
- Does y = −0.5x² + 2 have a max or min? Value?
- y-intercept of y = 3x² − x + 7?
Answers: 1) yes (second differences 2) 2) (−2, −3) 3) maximum, 2 4) 7
Quick recap
- A quadratic function graphs as a parabola.
- Second differences are constant.
- The vertex, axis of symmetry and optimal value describe its shape.