A quadratic function in one variable is a polynomial function of degree 2. It has the standard form y equals a x squared plus b x plus c, where a, b, and c are constants, and the crucial condition is that a cannot equal zero. This creates a parabolic curve when graphed.
The quadratic function has three main components. The coefficient a determines the direction and width of the parabola. The coefficient b affects the horizontal position of the vertex. The constant c represents the y-intercept. For example, in y equals 2x squared minus 3x plus 1, we have a equals 2, b equals negative 3, and c equals 1.
The coefficient 'a' has a significant impact on the parabola's shape. When a is positive, the parabola opens upward. When a is negative, it opens downward. The absolute value of a affects the width: larger values make the parabola narrower, while smaller values make it wider. Let's see how different values of a change the graph.
Every quadratic function has a vertex, which is the highest or lowest point on the parabola. The vertex x-coordinate is found using the formula x equals negative b divided by 2a. The axis of symmetry is a vertical line passing through the vertex. For the example y equals x squared minus 4x plus 3, the vertex is at (2, -1) and the axis of symmetry is x equals 2.
Quadratic functions have many real-world applications. They model projectile motion in physics, optimization problems in engineering, and profit maximization in economics. For example, a ball thrown upward follows a parabolic path described by h equals negative 16t squared plus 64t plus 6, where h is height and t is time. The ball reaches its maximum height of 70 feet at 2 seconds.