# How do we find the range of the equation f(x)=-x^2-10xI already know how to find the domain of the equation, I'm just having trouble trying to figure out the range.

Asked on by katz326

mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on

`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:

`x=-(b)/(2a)`

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`

`y=-(-5)^2-10(-5)=25`

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:

`y<=25`

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