how do you find intersections between two graphical functions?

When two lines intersect, the point of intersection will be a finite set of coordinates which can be obtained from the equation of either line. A sketch of the the functions will help you identify how many points of intersection you need to work out. As the y and x value are the same in both equations at a point of intersection, it is possible to make either x or y the subject of both equations (if the function is a quadratic equation it is far more beneficial to make y the subject). This means that the two equations are equal to each other and so an expression for the object of the equation can be obtained. From here the expression needs to be solved. If both functions are linear x can be found by re arranging the equation to make x the subject. If at least one of the equations is quadratic more advanced techniques may be needed to find the values of x at the points of intersection. If there is only an x² term and no x term then you will need to re-arrange the equation to make x² the subject before taking the square root (remembering that a square root will produce a positive and negative value for x). If an x term is present then you will need to collect all the terms onto one side of the equation meaning the expression will be equal to zero. At this point it may be possible to factorise the expression to obtain the two values for x but if not the quadratic formula can be applied. Once the value(s) of x have been obtained, sub the value(s) back into one of the original equations to obtain a value for and then check that the same y value is retrieved when subbing the x value back into the second original equation.