What is the domain and range of #y=4-x^2#?

1 Answer
Jun 22, 2018

Domain: All Real Numbers
Range: [4, -#oo#)


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.

  1. The parabola's vertex is at (0,4), as, in the parabola's equation,
    h = 0 and k = 4.
  2. 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.