F(x) = 5 + 3x − x2 what is the range

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].

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