Please use " ^ " to indicate exponentiation: F(x) = 5 + 3x − x^2. One way to determine the range of a quadratic function, such as this function is, is to find the vertex. The y-value of the vertex is the max or the min of the function.
In this case, we have f(x) = -x^2 + 3x + 5, and the associated coefficients are a = -1, b = 3 and c = 5. The axis of symmetry is x = -b/[2a].
Here, the equation of the axis of symmetry is x = -3/[2*-1), or x = 3/2.
Find the corresponding y value by subbing 3/2 for x in f(x) = -x^2 + 3x + 5:
f(3/2) = -(3/2)^2 + 3(3/2) + 5 = -9/4 + 9/2 + 20/4, or 9./4.
Thus, the vertex, representing a maximum, is (3/2, 29/4).
The range is (-infinity, 29/4].
