Polynomial functions with degree 2 are called quadratic functions. In this course, as is also true in general, we are primarily concerned with polynomial functions with real coefficients.
Quadratic Function
A quadratic function fis a function of the form
f(x)=ax2+bx+c
where a,b,and care complex numbers, with a≠0.
When we only consider quadratic functions with real coefficients, the graphs of these functions will be parabolas opening either up or down, depending upon the sign of a(with the parabola opening upward for a>0,and downward for a<0.) All of these parabolas have a vertical axis, called the axis or axis of symmetry, around which the graph is symmetric. The equation for the axis of symmetry of a parabola with equation f(x)=a(x−h)2+kis x=h. The vertex is at (h,k).If the equation for a quadratic function is given in the form f(x)=ax2+bx+c it can be converted into the form f(x)=a(x−h)2+kby competing the square, or by using h=−b2aand k=f(h).
In general, here is a review of the characteristics of the graph of f(x)=a(x−h)2+k.
Domain and Range for a Quadratic Function with Real Coefficients
Since the graph looks like either a parabola that opens up or one that opens down, the domain will always be (−∞,∞).The range will depend on the sign of the leading coefficient, a,in such a way that:
For a>0andf(x)=a(x−h)2+k,therangeis[k,∞)
For a<0andf(x)=a(x−h)2+k,therangeis(−∞,k]
Polynomial Functions Examples with Quadratic Functions
a>0andf(x)=a(x−h)2+k,therangeis[k,∞)
a<0andf(x)=a(x−h)2+k,therangeis(−∞,k]