Question

Find the area of a quadrilateral formed by the points of intersection of the graphs f(x) = (x − 3)(x + 1) and g(x) = 4x − 2x^2 + 6 and the points of extrema of f(x) and g(x).

Answer 1

24

Explanation:

The extrema is halfway between the roots, so for f(x), it has coordinates (1, f(1)) = (1, -4).

For g(x), we can rewrite it as -2(x²-2x-3) = -2(x-3)(x+1). We can see that the extrema has coordinates (1, 8).

Finding the intersection,

`(x-3)(x+1)=-2(x-3)(x+1) \\ \\ 3(x-3)(x+1)=0 \\ \\ x=-1, 3 \\ \\ x=-1 \implies f(x)=g(x)=0 \\ \\ x=3 \implies f(x)=g(x)=0`

So, we need to find the quadrilateral with vertices (1,-4), (1,8), (-1,0), and (3,0).

We observe this is a kite with diagonals of lengths 4 and 12. So, using the formula for the area of a kite, the area is 24.