Definition: A quadratic function is a function that can be written as f(x) = ax^2 + bx + c with a not equal to 0, and its graph is a parabola.
What defines a quadratic function?
The highest power of x is 2. The letter a controls the opening: positive a opens upward, negative a opens downward. The constant c is the y-intercept.
What features should you know?
The vertex is the turning point, the axis of symmetry is the vertical line through it, and the zeros are where the graph meets the x-axis. The x-coordinate of the vertex is -b / (2a).
Example. For f(x) = 2x2 – 12x + 7, the vertex has x = -(-12) / (2 · 2) = 3. Then f(3) = 18 – 36 + 7 = -11. The vertex is (3, -11) and, because a = 2 is positive, it is the minimum point. The y-intercept is 7.
Which errors are frequent?
Using b / (2a) without the minus sign, or forgetting to substitute back to get the y-coordinate. A quadratic function is not linear: equal steps in x do not give equal steps in y. The second differences of its outputs are constant, which is a handy test with tables.