A quadratic function is any polynomial function where the highest power of the variable is 2. Its standard form is f(x) = ax² + bx + c, where a, b, and c are real numbers and a is not zero. That single requirement, a squared term with a nonzero coefficient, is what separates a quadratic from every other type of function. If a were zero, the x² term would disappear and you’d be left with a linear function.
The Three Parts of Standard Form
In f(x) = ax² + bx + c, each piece controls something specific about the function’s behavior.
The coefficient a (the number in front of x²) is the most important. It determines whether the graph opens upward or downward and how wide or narrow the curve is. A positive a opens upward like a cup; a negative a opens downward like an arch. A larger absolute value of a makes the curve narrower, while a smaller one makes it wider.
The coefficient b influences where the curve shifts horizontally. It works together with a to determine the location of the vertex and the axis of symmetry.
The constant c is the y-intercept. When you plug in x = 0, the x² and x terms both vanish, and you’re left with f(0) = c. So the graph always crosses the vertical axis at the point (0, c).
The Parabola: Shape of Every Quadratic
Every quadratic function produces a U-shaped curve called a parabola. This shape is symmetric, meaning one half is a mirror image of the other. The vertical line that divides the parabola into two equal halves is called the axis of symmetry, and its equation is x = -b / 2a.
The turning point of the parabola is the vertex. It sits right on the axis of symmetry, so its x-coordinate is also -b / 2a. You find the y-coordinate by plugging that x value back into the original function. The vertex represents either the minimum value of the function (when the parabola opens upward) or the maximum value (when it opens downward).
Vertex Form: A Different Way to Write It
Quadratic functions can also be written as f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to read the vertex directly from the equation without any calculation. The value h tells you how far left or right the parabola has shifted from the origin, and k tells you how far up or down.
For example, f(x) = 2(x – 3)² + 5 has its vertex at (3, 5), opens upward because a is positive, and is narrower than a basic x² parabola because a is greater than 1. You can always convert between standard form and vertex form by completing the square or expanding the terms.
How Quadratics Cross the X-Axis
The x-intercepts of a quadratic (also called roots or zeros) are the points where the function’s output equals zero. You find them by solving ax² + bx + c = 0, and the most reliable method is the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a.
The expression under the square root, b² – 4ac, is called the discriminant. It tells you how many real solutions exist before you finish solving:
- Positive discriminant: two distinct x-intercepts, meaning the parabola crosses the x-axis in two places.
- Zero discriminant: exactly one x-intercept, meaning the vertex just touches the x-axis.
- Negative discriminant: no real x-intercepts, meaning the entire parabola floats above or below the x-axis without ever crossing it.
The y-intercept is simpler. Evaluate f(0), and you get c directly. So for f(x) = 3x² + 5x – 2, the y-intercept is -2.
Why Quadratics Show Up Everywhere
Quadratic functions model any situation where something accelerates, peaks, and then reverses. The most classic example is projectile motion. When you throw a ball into the air, gravity pulls it back down at a constant rate of acceleration, and the height over time follows a parabolic path.
A baseball hit straight up at 140 feet per second from a height of 4 feet can be modeled by h(t) = -16t² + 140t + 4. The coefficient -16 comes from gravitational acceleration (in feet per second squared, halved). The vertex of this parabola gives both the time at which the ball reaches its peak and the maximum height. Setting h(t) = 0 and solving tells you when the ball hits the ground.
Beyond physics, quadratics appear in business (maximizing profit or minimizing cost), engineering (designing arches and cables), and biology (modeling population growth over short intervals). Any time a relationship involves a quantity being squared, a quadratic function is likely at work.
Quick Test: Is It Quadratic?
To determine whether a function is quadratic, check two things. First, the highest exponent on the variable must be exactly 2. If you see x³ or higher, it’s a higher-degree polynomial. If the highest power is x¹, it’s linear. Second, the coefficient on the x² term cannot be zero. An equation like 0x² + 4x + 1 is just 4x + 1, which is linear.
Functions that might not look quadratic at first glance sometimes are. For instance, f(x) = (x + 3)(x – 1) doesn’t have an obvious x² term, but multiplying it out gives f(x) = x² + 2x – 3, which fits standard form with a = 1, b = 2, and c = -3. Similarly, f(x) = 5(x – 4)² + 7 is already in vertex form and clearly quadratic because expanding it produces a leading x² term with a nonzero coefficient.