What Are the Points of Intersection of f and g Given f(x) 2x^2 - 6 and g(x) -2x 6?

To find the points of intersection of the functions f(x) 2x^2 - 6 and g(x) -2x 6, we need to set them equal to each other and solve for x. This involves the following steps:

Setting the Functions Equal to Each Other

Starting with the equation:

[ 2x^2 - 6 -2x 6 ]

Rearranging the equation to isolate terms on one side:

[ 2x^2 2x - 12 0 ]

Simplifying the Equation

We can simplify this equation by dividing all terms by 2:

[ x^2 x - 6 0 ]

Factoring the Quadratic Equation

This quadratic equation can be factored into:

[ (x 3)(x - 2) 0 ]

Solving for x

Setting each factor to zero to find the values of x:

[ x 3 0 quad Rightarrow quad x -3 ]

And

[ x - 2 0 quad Rightarrow quad x 2 ]

Finding the Corresponding y-Coordinates

To find the y-coordinates of the points of intersection, we substitute these x-values back into the function g(x) (since it is simpler):

For x -3:

[ g(-3) -2(-3) 6 6 6 12 ]

For x 2:

[ g(2) -2(2) 6 -4 6 2 ]

The Points of Intersection

Thus, the points of intersection are:

(-3, 12) (2, 2)

Summary

The points of intersection of the functions f(x) 2x^2 - 6 and g(x) -2x 6 are (-3, 12) and (2, 2).

Confirming the Intersection on a Graph

The points of intersection can be confirmed by plotting the functions and observing where they cross. By solving the equation and substituting back, we find that the intersection points are exactly at (-3, 12) and (2, 2).

Additional Context and Tools

Newton's method, a numerical method for finding roots of equations, can also be used to find approximate solutions. The R script provided can be used to plot the functions and estimate the roots accurately.

For a more in-depth understanding, consider using tools like WolframAlpha, which can solve the equation and provide visual confirmation of the intersection points.