SOLUTION: 4x^2+40x+97 tell me the vertex and line of symmetry

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Question 79924: 4x^2+40x+97 tell me the vertex and line of symmetry
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Check out this solver for more problems dealing with the vertex and axis of symmetry.
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=4+x%5E2%2B40+x%2B97 Start with the given equation



y-97=4+x%5E2%2B40+x Subtract 97 from both sides



y-97=4%28x%5E2%2B10x%29 Factor out the leading coefficient 4



Take half of the x coefficient 10 to get 5 (ie %281%2F2%29%2810%29=5).


Now square 5 to get 25 (ie %285%29%5E2=%285%29%285%29=25)





y-97=4%28x%5E2%2B10x%2B25-25%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25 does not change the equation




y-97=4%28%28x%2B5%29%5E2-25%29 Now factor x%5E2%2B10x%2B25 to get %28x%2B5%29%5E2



y-97=4%28x%2B5%29%5E2-4%2825%29 Distribute



y-97=4%28x%2B5%29%5E2-100 Multiply



y=4%28x%2B5%29%5E2-100%2B97 Now add 97 to both sides to isolate y



y=4%28x%2B5%29%5E2-3 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=4, h=-5, 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=4x%5E2%2B40x%2B97 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B40x%2B97%29 Graph of y=4x%5E2%2B40x%2B97. Notice how the vertex is (-5,-3).



Notice if we graph the final equation y=4%28x%2B5%29%5E2-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4%28x%2B5%29%5E2-3%29 Graph of y=4%28x%2B5%29%5E2-3. Notice how the vertex is also (-5,-3).



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