A quadratic function is a polynomial function of degree 2. Its general form is y equals a x squared plus b x plus c, where a, b, and c are constants, and a cannot be zero. The graph of a quadratic function is always a parabola, which has a distinctive U-shape or inverted U-shape.
The coefficient 'a' plays a crucial role in determining the shape and orientation of the parabola. When 'a' is positive, the parabola opens upward like a U-shape. When 'a' is negative, it opens downward like an inverted U. The absolute value of 'a' affects the width: larger values make the parabola narrower, while smaller values make it wider.
Every parabola has two key features: the vertex and the axis of symmetry. The vertex is the highest or lowest point of the parabola, depending on whether it opens upward or downward. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For any quadratic function y equals a x squared plus b x plus c, the x-coordinate of the vertex can be found using the formula x equals negative b over 2a.
Quadratic functions can be written in three different forms, each highlighting different properties. The standard form y equals a x squared plus b x plus c is most common. The vertex form y equals a times x minus h squared plus k makes it easy to identify the vertex at coordinates h, k. The factored form y equals a times x minus r1 times x minus r2 clearly shows the x-intercepts or roots at r1 and r2. Each form is useful for different purposes in solving problems.
Quadratic functions have numerous real-world applications. In physics, they describe projectile motion, such as the path of a thrown ball. In economics, they model profit optimization problems. Engineers use them in bridge design and architecture for parabolic arches. The example shown demonstrates a projectile's height over time, where the quadratic function helps us find the maximum height and determine when the object will land. Understanding quadratic functions is essential for solving many optimization problems in science, engineering, and everyday life.