Solving Nonlinear Equations
In the following example, we find the solutions to a quadratic function
by graphing the function and then finding the x-intercepts.
Example
Given the function: f(x) = x2 + 8x +12
a. Graph the function.
b. Use the graph to find the solutions to x2 + 8x +12
= 0.
Solution
a. The graph of the function f(x) = x2 + 8x +12
is a parabola since it has
the form y = ax2 + bx + c. Here, a = 1, b = 8, and c = 12.
To graph the parabola, first find the x-coordinate of the vertex and then
calculate ordered pairs on either side of the vertex.
| Here is the formula for the
x-coordinate 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 x-coordinate 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 = x2 + 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 x-axis at (-6, 0) and (-2, 0).
So, the solutions of x2 + 8x +12 = 0 are x = -6 and x = -2.
|
