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 Dependent Variable

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# 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 50 -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.