What is the greatest integer value of b such that -4 is not in the range of ?

Thank you!!

Guest Jul 18, 2017

#2**+2 **

Since the parabola turns upward, we re looking for the y value of a vertex that is **>** - 4

So....we can solve the following to find the x coordinate of this vertex

-b/ (2 * a ) = x coord of the vertex

-b / (2 * 1) = x

-b/2 = x

And we require that

x^2 + bx + 12 > - 4 substituting, we have that

(-b/2)^2 + b (-b/2) + 12 > -4

b^2 /4 -b^2/2 > -16

-b^2 / 4 > -16

-b^2 > -64 multiply by -1 and reverse the inequality sign

b^2 < 64

So

-8 < b < 8 will produce a parabola whose range > - 4

And.....the graph is still in range whenever the largest integer value of b = 7

For comparative purposes......see the graphs here for some different values of "b"

https://www.desmos.com/calculator/my77qzagdu

Note that when l b l < 8 the range is > -4

But when l b l ≥ 8, the graph is out of range

CPhill
Jul 18, 2017

#2**+2 **

Best Answer

Since the parabola turns upward, we re looking for the y value of a vertex that is **>** - 4

So....we can solve the following to find the x coordinate of this vertex

-b/ (2 * a ) = x coord of the vertex

-b / (2 * 1) = x

-b/2 = x

And we require that

x^2 + bx + 12 > - 4 substituting, we have that

(-b/2)^2 + b (-b/2) + 12 > -4

b^2 /4 -b^2/2 > -16

-b^2 / 4 > -16

-b^2 > -64 multiply by -1 and reverse the inequality sign

b^2 < 64

So

-8 < b < 8 will produce a parabola whose range > - 4

And.....the graph is still in range whenever the largest integer value of b = 7

For comparative purposes......see the graphs here for some different values of "b"

https://www.desmos.com/calculator/my77qzagdu

Note that when l b l < 8 the range is > -4

But when l b l ≥ 8, the graph is out of range

CPhill
Jul 18, 2017