SOLUTION: Does the function have a minimum or maximum value. where does the minimum or maximum value occur what is the functions minimum or maximum value f(x)=-2x^2-16x-35

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Question 645273: Does the function have a minimum or maximum value.
where does the minimum or maximum value occur
what is the functions minimum or maximum value

f(x)=-2x^2-16x-35

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=-2x%5E2-16x-35+
1.
since -2 is less than zero, the parabola opens downward and has a global maximum value
2. complete the square
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-2+x%5E2-16+x-35 Start with the given equation



y%2B35=-2+x%5E2-16+x Add 35 to both sides



y%2B35=-2%28x%5E2%2B8x%29 Factor out the leading coefficient -2



Take half of the x coefficient 8 to get 4 (ie %281%2F2%29%288%29=4).


Now square 4 to get 16 (ie %284%29%5E2=%284%29%284%29=16)





y%2B35=-2%28x%5E2%2B8x%2B16-16%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 16 does not change the equation




y%2B35=-2%28%28x%2B4%29%5E2-16%29 Now factor x%5E2%2B8x%2B16 to get %28x%2B4%29%5E2



y%2B35=-2%28x%2B4%29%5E2%2B2%2816%29 Distribute



y%2B35=-2%28x%2B4%29%5E2%2B32 Multiply



y=-2%28x%2B4%29%5E2%2B32-35 Now add %2B35 to both sides to isolate y



y=-2%28x%2B4%29%5E2-3 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-2, h=-4, and k=-3. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-2x%5E2-16x-35 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%5E2-16x-35%29 Graph of y=-2x%5E2-16x-35. Notice how the vertex is (-4,-3).



Notice if we graph the final equation y=-2%28x%2B4%29%5E2-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2%28x%2B4%29%5E2-3%29 Graph of y=-2%28x%2B4%29%5E2-3. Notice how the vertex is also (-4,-3).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.






Notice how the vertex is at(-4,-3)-global maximum.