`f(x)=-x^2-10x` is a parabola opening down. Graphically:
The range is the possible y-values that can be obtained.
For an upside-down parabola, you can't get reach any y-value taller than the peak of the parabola. So what this question is really asking, is: how high is the peak of the parabola? (the "vertex")
So: we want the y-coordinate of the vertex.
The x-coordinate of the vertex is given by the formula:
So for us, `x=-((-10)/(2(-1)))=-5`
To get the y-coordinate, plug the x-coordinate into the original function: `y=-x^2-10x`
So the vertex of the parabola is at (-5,25)
So the highest possible y value is 25. We can get every y value less than or equal to 25, but we can't reach any higher than that.
That is, if you pick any number `<=25` , there is a spot somewhere on the parabola with that number as its y-value. But if you pick a number `>25` , you can't find any spot on the parabola with that as the y-value.
So the range is: