Function library
Quadratic Function
f(x) = x^2 - 4
The parabola x² − 4: a U-shaped curve with roots at ±2 and its lowest point at (0, −4).
The quadratic function f(x) = x² − 4 is the simplest parabola with something interesting going on: it crosses the x-axis twice, dips below it, and turns around at a single lowest point. Squaring makes every input non-negative, so x² is smallest at x = 0, and subtracting 4 slides the whole U-shape down four units. The result is a symmetric curve with vertex at (0, −4), opening upward forever.
Quadratics are the workhorses of algebra: they model anything where a quantity depends on the square of another — the area of a square, the height of a thrown ball over time, the profit of a business with linear demand. The example x² − 4 is especially instructive because it factors cleanly as (x − 2)(x + 2), so its x-intercepts at 2 and −2 can be read straight off the algebra. Plot it in the calculator and watch the symmetry about the y-axis.
Computed properties
Calculated by the graphing calculator's own math engine at build time for x^2 - 4.
- Roots in [−10, 10]
- -2.0000, 2.0000
- y-intercept f(0)
- -4.0000
- Local extrema in [−10, 10]
-
- local minimum at (0.0000, -4.0000)
- Sample values
- f(-2) = 0 · f(-1) = -3 · f(0) = -4 · f(1) = -3 · f(2) = 0
- Derivative f′(1)
- 2.0000
- Integral ∫₀¹ f(x) dx
- -3.6667
Key facts
- Factored form: x² − 4 = (x − 2)(x + 2); roots (x-intercepts) at x = 2 and x = −2.
- Vertex (global minimum) at (0, −4); axis of symmetry is the y-axis (x = 0).
- Domain: all real numbers; range: y ≥ −4.
- Even function: f(−x) = f(x); the graph mirrors across the y-axis.
- y-intercept at (0, −4); the function is negative between the roots and positive outside them.
- Derivative f′(x) = 2x; the slope is zero at the vertex and the second derivative is the constant 2.
What this quadratic is
Every quadratic has the form ax² + bx + c, and its graph is always a parabola — the U-shape you get from squaring. Here a = 1 (positive, so the U opens upward), b = 0 (no tilt, so the vertex sits on the y-axis), and c = −4 (the y-intercept). The vertex formula x = −b/(2a) gives x = 0, and f(0) = −4, confirming the minimum at (0, −4).
Factoring reveals the roots: x² − 4 = (x − 2)(x + 2), a difference of squares, so the curve crosses the x-axis exactly where each factor is zero — at x = 2 and x = −2. Between the roots the function is negative (the dip below the axis); outside them it is positive and grows without bound. Because the x² term dominates for large |x|, both arms of the parabola head to +∞.
Symmetry, vertex, and rate of change
The y-axis is the parabola’s axis of symmetry: f(−x) = f(x), so the left half mirrors the right. The vertex is the parabola’s turning point — here the global minimum, since the arms rise forever. Any quadratic has exactly one vertex and exactly one extreme value, which is why quadratics are the go-to model for optimization: maximum profit, minimum cost, highest point of a trajectory.
The derivative f′(x) = 2x tells the rest of the story: negative for x < 0 (falling into the vertex), zero at x = 0 (the flat bottom), positive for x > 0 (climbing out). The slope itself grows linearly, a constant second derivative of 2 — the signature of constant acceleration, which is why distance under gravity is quadratic in time.
Where quadratics appear
Throw a ball and its height follows a parabola: h(t) = −4.9t² + v₀t + h₀, the same shape as x² − 4 but flipped and shifted. Areas and volumes produce quadratics and cubics naturally — doubling a square’s side quadruples its area — and the quadratic formula solves every equation of this type, including this one: x = ±√4 = ±2.
In economics, profit as a function of price is often modeled as a downward-opening parabola (revenue rises then falls as price climbs), and its vertex gives the optimal price. In statistics, least-squares fitting minimizes a quadratic error function, and the normal distribution’s bell curve is e^(−x²) — a quadratic in the exponent.
Frequently asked questions
How do you find the roots of x² − 4?
Factor it as a difference of squares: x² − 4 = (x − 2)(x + 2). A product is zero when any factor is zero, so x = 2 or x = −2. Equivalently, the quadratic formula gives x = (0 ± √(0 + 16))/2 = ±2.
What is the minimum value of x² − 4?
−4, attained at x = 0. Since x² ≥ 0 for all real x, subtracting 4 gives x² − 4 ≥ −4, with equality only when x² = 0. The vertex (0, −4) is the parabola’s lowest point, and the function increases without bound on both sides.
Why is the graph symmetric?
Because only even powers of x appear: (−x)² − 4 = x² − 4, so f(−x) = f(x). Every input and its negative give the same output, which mirrors the right half of the graph across the y-axis onto the left half.