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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

Best Answer 

 #2
avatar+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

 

 

cool cool cool

 Jul 18, 2017
edited by CPhill  Jul 18, 2017
edited by CPhill  Jul 18, 2017
 #1
avatar+27547 
0

Oops!  See CPhill's reply.

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

 

 

cool cool cool

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

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