Let's divide the shaded region into two areas:
area 1: x = 0 ---> x = 2
ares 2: x = 2 ---> x = 4
In area 1, we need to find the area under g(x) = x and in area 2, we need to find the area between g(x) = x and f(x) = (x - 2)^2. Now let's set up the integrals needed to find the areas.
Area 1:
Area 2:
Therefore, the area of the shaded portion of the graph is
A = A1 + A2 = 5.34
We have to find the parameter a so that (-1,2) is part of the function f(x) = ax²+4.
To check if a point is part of a function, we can replace the values of x and y = f(x) with the coordinates of the point and then, if the equation stays true, then the point is part of the function.
So for (x,y) = (-1,2) to be part of the function y = f(x), this equation has to stands true:
Then, the function would have to be f(x) = -2x² + 4.
We can check with a graph as:
Answer: a = -2
