__Category__: Analytic Geometry, Algebra"Published in Suisun City, California, USA"

Find the coordinates of the points of intersection for

__Solution__:

To illustrate the problem, let's draw the graph of the two given equations above.

For parabola, we need to reduce into standard form as follows

The center of parabola is C(0, -2) with vertex V(0, -1¾), and the ends of the latus rectum L

_{1}(½, 0) and L

_{2}(-½, 0). The curve of a parabola opens upward because x

^{2}is positive.

For a line, we need to express into slope-intercept form as follows

The slope of a line is 2 and the y-intercept is 1.

From the detailed information of a parabola and a line, we can sketch the graphs as follows

Photo by Math Principles in Everyday Life |

As you can see that there are two points of intersection for a parabola and a line. To get the coordinates of their points of intersection, let's solve for the systems of the given two equations as follows

Change the sign of the second equation and then proceed with the addition as follows

Equate each factor to zero and solve for the value of x

For x - 3 = 0 For x + 1 = 0

x = 3 x = -1

Next, substitute the values of x to either one of the given two equations in order to get the value of y. In this case, we will use the equation of a line so that it is easier to get the value of y as follows

If x = 3, then y will be

If x = -1, then y will be

Therefore, their points of intersection are

**P**and

_{1}(3, 7)**P**.

_{2}(-1, -1)