Solving Nonlinear Equations
In the following example, we find the solutions to a quadratic function
by graphing the function and then finding the xintercepts.
Example
Given the function: f(x) = x^{2} + 8x +12
a. Graph the function.
b. Use the graph to find the solutions to x^{2} + 8x +12
= 0.
Solution
a. The graph of the function f(x) = x^{2} + 8x +12
is a parabola since it has
the form y = ax^{2} + bx + c. Here, a = 1, b = 8, and c = 12.
To graph the parabola, first find the xcoordinate of the vertex and then
calculate ordered pairs on either side of the vertex.
Here is the formula for the
xcoordinate of the vertex.

x 

Substitute a = 1 and b = 8. 
x 

Simplify. 

= 4 
Now, make a table of values by choosing values of x on either side of
the xcoordinate of the vertex, x = 4.
x 
y 
1 2
3
4
5
6
7 
5 0
3
4
3
0
5 
Finally, use the table to graph y = x^{2} + 8x +12.
Note:
The line x = 4 is the axis of symmetry of
the parabola.
That is, if you fold the graph along the line
x = 4 one side of the graph will lie on
top of the other.
b. The graph crosses the xaxis at (6, 0) and (2, 0).
So, the solutions of x^{2} + 8x +12 = 0 are x = 6 and x = 2.
