Final answer:
The function that expresses the area of a rectangle with base x and perimeter 2a is A(x) = x². We derive this by using the perimeter formula to express length in terms of a and x, and then apply the area formula of length times width.
Step-by-step explanation:
To determine a function that expresses the area of a rectangle with base x and perimeter 2a, we need to use the formula for the perimeter of a rectangle, which is P = 2l + 2w, where l is the length and w is the width (or base in this case).
Since the perimeter is given as 2a, and the base as x, we can set up the equation 2a = 2l + 2x. Simplifying this gives us a = l + x. To find the length l, we rearrange to l = a - x. The area of a rectangle is length × width, so substituting l with a - x we get A(x) = x(a - x). Now we expand this to get A(x) = ax - x². To find the quadratic function in terms of x alone, we need the coefficient of x to be zero since a is a constant. This can be achieved when a = 2x, which gives us the function A(x) = x(2x - x) = x².