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# I need some help!

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

Thank you!!

Jul 18, 2017

#2
+98130
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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

Jul 18, 2017
edited by CPhill  Jul 18, 2017
edited by CPhill  Jul 18, 2017

#1
+27547
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Oops!  See CPhill's reply.

Jul 18, 2017
edited by Alan  Jul 19, 2017
#2
+98130
+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
edited by CPhill  Jul 18, 2017
edited by CPhill  Jul 18, 2017