Summary
Highlights
A quadratic relation is defined by the equation y = ax^2 + bx + c, where 'a', 'b', and 'c' are real numbers and 'a' is not zero. Unlike linear graphs, quadratic relations graph as U-shaped curves called parabolas. These parabolas can open upwards or downwards, vary in width, and are not necessarily centered at the origin.
The simplest quadratic relation is y = x^2, where a=1 and b=c=0. Plotting points shows its characteristic parabolic curve. Plugging in both positive and negative x-values results in the same positive y-values due to squaring, creating a symmetrical graph.
The vertex is the point on the parabola where the curve changes direction. If the parabola opens up, the vertex is the minimum point; if it opens down, it's the maximum point. A parabola is symmetrical, and the imaginary vertical line that divides it into two symmetrical halves is called the axis of symmetry.
A key way to identify a quadratic relation is by examining the second differences of its y-values when the x-values are evenly spaced. By creating a table with x, y, first difference (difference between consecutive y-values), and second difference (difference between consecutive first differences), a quadratic relation will always have constant second differences.
Linear equations never have an exponent of two on any variable. In contrast, quadratic equations always include an x^2 term, which is the defining characteristic that distinguishes them from linear equations. This is a quick visual cue to identify a quadratic equation.