The Quadratic Function


The quadratic function has the form:

F(x)  =  y  =    a  +  bx  +  cx2

where a, b, and c are numerical constants and c is not equal to zero.


Note that if c were zero, the function would be linear.


An advantage of this notation is that it can easily be generalized by adding more terms.  We could for example write equations such as

y  =    a  +  bx  +  cx2  + dx3

y  =    a  +  bx  +  cx2  + dx3  + ex4


The quadratic formula

In many books quadratic equations are written as

ax+ bx  +  c =  0

In this case the quadratic formula is given by

Note that the denominator is then 2a instead of 2c.


Some common examples of the quadratic function


Notice that the graph of the quadratic function is a parabola.  This means it is a curve with a single bump. The graph is symmetric about a line called the axis of symmetry.  The point where the axis of symmetry intersects the parabola is known as the vertex.










Graphing the quadratic function

            Construct a table with values of x and f(x).

            Plot the data points.

            Connect the data points with a smooth line.

            This is easily done with Excel.



Example 1         f(x)   = 12 - 8x  +x2






Example 2         f(x)   = -4 + 5x  -x2




The quadratic formula, an example.


In general the supply of a commodity increases with price and the demand decreases.  The market for the commodity is in equilibrium when supply equals demand.


In this example we are considering two functions of the same independent variable, price.  We want to find the equilibrium price and the corresponding demand.


The supply function is a quadratic equation given by          S(p)  =  2p  +  4p2


The demand function is a linear function given by               D(p)  =  231  -  18p


To find the intersection of the two curves set supply equal to demand and solve for p.


            S(p)  =  2p  +  4p2   =  231  -  18p  =  D(p)


After collecting terms we obtain the quadratic equation


            231  -  20p  -4p2  =  0


Note that  this has the form of the quadratic equation


            y  =    a  +  bx  +  cx2


            0  =   231 -20p  -4p2


Solve the equation by means of the quadratic formula  where a = 231, b = -20, and c = -4.









Since price cannot be negative the value of -10.5 can be eliminated.  Supply will equal demand when the price is $5.50.  At that price it is possible to find the corresponding demand and supply.


D(5.50)  =  231  -  18(5.50)       =  132

S(5.50)  =  2(5.50)  +  4(5.50=   132