Quadratic function is a function whose function value changes in proportion to the square of the independent variable.
A quadratic function on a set of real numbers is any function f that is given by the formula:
f: y=ax2 +bx+c
where a, b, c∈R ∧ a ≠ 0. The domain is all real numbers Dy = R.
The graph of a quadratic function is a curve called parabola. The important points of a parabola are its vertex V and the intersections of the parabola with the x and y coordinate axes. The graph is symmetric along the axis of the parabola, which is parallel to the y-axis and passes through its vertex.
V – the vertex of the parabola
X1, X2 – intersections with the x-axis
Y – intersection with the y-axis
Influence of coefficients a, b, c on a graph of quadratic function.
Influence of coefficient a
For a positive coefficient a the graph of the quadratic function is convex („open“ upwards), for a negative coefficient a the graph of the quadratic function is concave („open“ downwards). With increasing absolute value of the coefficient a, the parabola is more „attached“ to its axis.
Influence of coefficient b
The coefficient b is related to the direction of the tangent to the parabola at its intersection with the y-axis. If we look at this tangent as another function, e.g. g, then the prescription for this function is g: y = bx + c. If the coefficient b is zero, then the vertex of the parabola lies on the y-axis.
Influence of coefficient c
The graph of the function intersects the y-axis at a point with a ypsilon coordinate equal to the coefficient c.
Calculation of the intersections of a graph of a quadratic function y=ax2 +bx+c with the x-axis
Each point lying on the x-axis has a ypsilon coordinate equal to zero, after substituting y = 0 into the formula of the quadratic function we obtain a quadratic equation, and by solving it we obtain the required intersections:
In special cases, the decomposition of a quadratic trinomial into the product of root factors can be used to determine the coordinates of intersections.
Calculation of the coordinates of vertices of the parabola of a quadratic function y=ax2 +bx+c
In general, we obtain the coordinates of the vertex easily using the formula:
these formulas are obtained by so-called completing the square of a function and several algebraic modifications. By completing the square of a specific given function, we obtain specific coordinates of the vertex even without knowing the formulas. You can also use the intersections of a parabola with the x-axis or the first derivative of a function to find the coordinates of its vertex.