SOLUTION: What must be the value of b so that the graph of {{{y = 2x^2 + bx + 4}}} has a range of {y|y>=3}

Algebra ->  Graphs -> SOLUTION: What must be the value of b so that the graph of {{{y = 2x^2 + bx + 4}}} has a range of {y|y>=3}      Log On


   

Question 1049701: What must be the value of b so that the graph of y+=+2x%5E2+%2B+bx+%2B+4 has a range of {y|y>=3}
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Put into standard form using completing the square. The equation in that form allows you to read the minimum value of the range directly in the standard form.

y=2%28x%5E2%2B%28b%2F2%29x%2B2%29

2%28x%5E2%2B%28b%2F2%29x%2B%28b%2F4%29%5E2%2B2-%28b%2F4%29%5E2%29

2%28%28x%2Bb%2F4%29%5E2%2B2-b%5E2%2F16%29

2%28%28x%2Bb%2F4%29%5E2%2B32%2F16-b%5E2%2F16%29

2%28%28x%2Bb%2F4%29%5E2%2B%2832-b%5E2%29%2F16%29

highlight_green%28y=2%28x%2Bb%2F4%29%5E2%2B%2832-b%5E2%29%2F8%29

The parabola has a minimum value vertex, and the parabola opens upward. You want range for y to be greater than or equal to 3, so set up the equation
%2832-b%5E2%29%2F8=3.

32-b%5E2=24
-b%5E2=24-32
b%5E2=32-24=8
highlight%28b=0%2B-+2%2Asqrt%282%29%29-----------Either of these will work. The vertical values are what you are interested in.