SOLUTION: what is the vertex form for:y=-1/3x^2+8/3x-25/3

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Question 684539: what is the vertex form for:y=-1/3x^2+8/3x-25/3
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
the vertex form:
y=a%28x-h%29%5E2%2Bk

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x-25%2F3 Start with the given equation



y%2B25%2F3=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x Add 25%2F3 to both sides



y%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%29 Factor out the leading coefficient %28-1%2F3%29



Take half of the x coefficient -8 to get -4 (ie %281%2F2%29%28-8%29=-4).


Now square -4 to get 16 (ie %28-4%29%5E2=%28-4%29%28-4%29=16)





y%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%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%2B25%2F3=%28-1%2F3%29%28%28x-4%29%5E2-16%29 Now factor x%5E2-8x%2B16 to get %28x-4%29%5E2



y%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2-%28-1%2F3%29%2816%29 Distribute



y%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3 Multiply



y=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3-25%2F3 Now add %2B25%2F3 to both sides to isolate y



y=%28-1%2F3%29%28x-4%29%5E2-3 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-1%2F3, 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=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3%29 Graph of y=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3. Notice how the vertex is (4,-3).



Notice if we graph the final equation y=%28-1%2F3%29%28x-4%29%5E2-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F3%29%28x-4%29%5E2-3%29 Graph of y=%28-1%2F3%29%28x-4%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.







so, (h,k)=(4,-3)