Lets begin by finding the domain of the function. As a parabola, and consequently a polynomial, the function 4-x^2 is defined for all real numbers – there is no point on its domain where the function is undefined. To better understand why the function is defined for all real numbers, see its graph: graph{4-x^2 [-10, 10, -5, 5]}
The range is also relatively simple to find.
y = 4 - x^2 can be re-written as y = -x^2+ 4, which has the form a(x- h )^2+ k , meaning that it is the vertex form of the parabola. This form tells us two important properties of the parabola.
- The parabola's vertex is at (0,4), as, in the parabola's equation,
h = 0 and k = 4.
- The parabola is concave down (it opens downwards) as a is negative.
As a result, the range of the parabola is all real values of y such that y<=4.
Were you to encounter a quadratic equation of a different form (such as, say, y=x^2+5x+6), you would need to either complete the square to derive the vertex form of that parabola or find the roots (x-intercepts) of the parabola, use those to find the midpoint between them (take the average of the two roots, and then find the y value of that point. Once you know the parabola's vertex and concavity, you can easily determine it's range, as shown above.
