How would you find the coordinates of the intersections of a graph with the x and y axes, and the coordinates of any turning points?
Take the graph y = x2 - 2x + 5 for example:
In order to find the intersection with the y axis, set the x values in the equation equal to 0 as this is where the graph would cross the y axis:y = (0)2 - 2(0) + 5 = 5, therefore the coordinate of the y intersection is (0, 5).
To find the x intersection, set y equal to 0: x2 - 2x + 5 = 0It is evident that the equation will have no solutions for x as the value of the discriminant b2 - 4ac = (-2)2 - (415) = -16 as it's a negative number therefore no solutions for x. If x did have solutions the discriminant would be above 0, if it equalled 0 it would have 1 solution, indicating the turning point on the x axis. To solve the quadratic if it did have solutions, factorise into double brackets and set each to 0 to find the solutions.
Finally onto the turning point(Alevel):find dy/dx by differentiating. set this equal to 0 as the turning point is where the gradient equals 0. Solve this equation for the x value, plug it back into the original equation to find y. y = x2 - 2x + 5 dy/dx = 2x - 2 2x - 2 = 0 2x = 2 x = 1 y = (1)2 -2(1) +5 = 4turning point = (1, 4)
