Find the range of a degree-2 polynomial function such as 2x^2 +1, or x^2 + 2x - 3.

There are 3 methods for finding the range of a polynomial fucntion. This is good news because if you arrive at the same answer using these methods you wil be more confident that it is the correct answer.The 1st method is to simply observe the behaviour of the function at different values of x. For example in 2x^2 + 1 (E1), after putting a few values of x in it is easy to see that the y value of the function (or the output) will always be positive. This makes sense, because if you square any real number and add 1, it's going to be positive. If you then simply try and find the minimum value, it is clear to see that setting x = 0 might be a good idea. Doing this, we get the range of f(x) is greater than or equal to 1.The second method to try is using a technique called completing the square, basically putting the same equation into a more useful format. So with x^2 + 2x - 3 (E2), we can rewrite this as (x+1)^2 - 4. This links to the first method in a way as we know the bracketed expression is always positive. Since this is true, the minium value of the bracket alone is 0, which arises when x = -1. Then we are left with 0 - 4 which gives -4. This is the minimum value of the function because if you try any other x values they will be greater than -4. Hence f(x) is equal to or greater than 4.The third method is to basically differentiate the function and set it equal to 0 to find the turning points. Differentiating (E2) we have 2x + 2 = 0. Remember that setting the differential equal to 0 tells us the minimum and maximum points of the function. Solving for x we get x = -1. This isn't the minimum value though, remember, this is when it occurs. So we need to plug x = -1 back into (E2). Doing this we get -4 as the minimum value, so again, the range of f(x) is that f(x) is greater than or equal to -4.It's also very helpful to draw the function out to make sure you get the minima/maxima the right way round.