MATH 120 Lecture Notes - Lecture 38: Quadratic Function
Dr. Matthew M. Conroy - University of Washington
1
Determining Quadratic Functions
A linear function, of the form f(x) = ax +b, is determined by two points. Given two points on
the graph of a linear function, we may find the slope of the line which is the function’s graph,
and then use the point-slope form to write the equation of the line.
A quadratic function, of the form f(x) = ax2+bx +c, is determined by three points. Given
three points on the graph of a quadratic function, we can work out the function by finding a, b
and calgebraically.
This will require solving a system of three equations in three unknowns. However, a general
solution method is not needed, since the equations all have a certain special form. In particular,
they all contain a +cterm, and this allows us to simplify to a two variable/two equation system
very quickly.
Here is an example.
Suppose we know f(x) = ax2+bx +cis a quadratic function and that f(−2) = 5,f(1) = 8,
and f(6) = 4. Note this is equivalent to saying that the points (−2,5),(1,8) and (6,4) lie on the
graph of f.
These three points give us the following equations:
5 = a((−2)2) + b(−2) + c= 4a−2b+c(1)
8 = a(12) + b(1) + c=a+b+c(2)
4 = a(62) + b(6) + c= 36a+ 6b+c(3)
Notice that +cterms dangling on the right-hand end of each equation.
By subtracting the equations in pairs, we eliminate the +cterm, and get two equations and two
unknowns.
We find, by subtracting equation (2) from equation (1), and by subtracting equation (2) from
equation (3), that
3a−3b=−3(4)
35a+ 5b=−4(5)
This system is then easily solved. We might, for example, simplify equation (4) to
b−a= 1
so that b=a+ 1, which when substituted into equation (5), yields
35a+ 5(a+ 1) = −4
which gives us
a=−
9
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Document Summary
A linear function, of the form f (x) = ax + b, is determined by two points. Given two points on the graph of a linear function, we may nd the slope of the line which is the function"s graph, and then use the point-slope form to write the equation of the line. A quadratic function, of the form f (x) = ax2 + bx + c, is determined by three points. Given three points on the graph of a quadratic function, we can work out the function by nding a, b and c algebraically. This will require solving a system of three equations in three unknowns. However, a general solution method is not needed, since the equations all have a certain special form. In particular, they all contain a +c term, and this allows us to simplify to a two variable/two equation system very quickly.
