What is the range of the equation y equals x2 plus 4?

Answer 1

This equation describes a parabola, so it's range on the x-axis will be infinite. To find it's vertex, we can take it's derivative and solve for zero:

y = x2 + 4

y' = 2x

Let y' = 0

0 = 2x

x = 0

Now we plug that x value into the original equation to find y:

y = 02 + 4

y = 4

So the vertex is at the point (0, 4). To see whether that's a minimum or a maximum, we need only take it's second derivative and check whether it's positive or negative at that point:

y' = 2x

y'' = 2

So the rate of change of the slope is positive, which means that the parabola's vertex is a minimum. We can say then that the equation has an infinite x range, and a y range from 4 to infinity.