What is a quadratic function?
A quadratic function, also known as second degree polynomial function, is a function of f: R → R defined by f(x) = ax² + bx + c, where a, b and c are real numbers and a ≠ 0.
f(x) = ax² + bx + c is also the standard form of quadratic functions.
How to graph a quadratic function?
The graph of a quadratic equation is a curve known as a parabola. To plot the parabola it is necessary to calculate important points such as the roots (or zeros) of the function, the vertex and the y-interception.
What are the roots of the quadratic function?
The roots (or zeros) of the quadratic function are the points where the graph intercepts the x-axis. To calculate the roots, it is necessary to write the function equal to zero, obtaining a second degree polynomial equation. There are many methods to solve second degree polynomial equations but, the most used one is the application of the Bhaskara formula, also known as the quadratic formula:
So, using the formula to solve the equation ax² + bx + c = 0, we get:
Note: the parabola intercepts the x-axis in, up to, two points.
- if b2-4ac > 0, the equation has two distinct real roots and the parabola intercepts the x-axis in two different points (x1 ≠ x2);
- if b2-4ac = 0, the equation has two equal real roots and the parabola is tangent to the x-axis (x1 = x2);
- if b2-4ac < 0, the equation has no real roots and the parabola does not intercept the x-axis;
Example of how to calculate the roots of a quadratic equation
Find the zeros of the function f(x) = x2 – 5x + 6.
a = 1
b = – 5
c = 6
x2 – 5x + 6 = 0
Replacing in the formula:
The value of b2-4ac > 0 , so there are two real distinct roots, which are 2 and 3, meaning that the function intercepts the x-axis at the points (2, 0) and (3, 0).
Where is the vertex of a quadratic function?
Another important point of the quadratic function is the vertex. The vertex represents the turning point of the line. Also, represents the highest value of the function, if the parabola is turned down, or, the lowest point, if the parabola is turned up.
To calculate the vertex coordinates the following expressions are to be used:
Using the example above, f(x) = x2 – 5x + 6:
So, the coordinates of the vertex are V = (xv , yv ) = (5/2 , – 1/4) or (2,5 , -0,25).
y-axis interception point
The parabola intercepts the y-axis at the value of the c coefficient. In the function above, the value of c = 6, therefore, the parabola intercepts the y-axis at the point (0, 6).
Graph of a quadratic function
As said before, the graph of a quadratic function is known as a parabola. Having calculated the roots, the vertex and the y-interception, one can now plot the graph. The graph of the function in the previous example is:
f(x) = x2 – 5x + 6
Plotting the graph of a quadratic function y = ax2 + bx + c, one will notice that:
- if a > 0 , the parabola has its concavity turned up;
- if a < 0 , the parabola has its concavity turned down;