How many intersection points can the graphs of the functions y = ax ^ 2 and y = bx ^ 3 have?

How many intersection points can the graphs of the functions y = ax ^ 2 and y = bx ^ 3 have? Here a and b are given coefficients. Determine the coordinates of these points at: 1) a = 2; b = -2; 2) a = 1/2; b = 1/3.

In order to analytically determine the intersection points of the graphs of functions, their equations must be equated.

In this case, we get:

a * x ^ 2 = b * x ^ 3;

Here you can reason in several ways.

If we take out the common factor, then we get:

x ^ 2 * (a – b * x) = 0;

From here we get two roots of the equation, respectively, two points.

If we take cubic and square functions in principle, then an equation of the third degree can have three roots, which means that the graphs will have three points of intersection.

1) 2 * x ^ 2 = -2 * x ^ 3;

x ^ 2 * (2 + 2 * x) = 0;

x = 0 and x = -1;

(0; 0) and (-1; 2) are intersection points.

2) 1/2 * x ^ 2 = 1/3 * x ^ 3;

x ^ 2 * (1/2 – 1/3 * x) = 0;

x = 0 and

1/2 = 1/3 * x;

3 = 2 * x;

x = 3/2;

(0; 0) and (3/2; 9/8) are the intersection points of the graphs.