SOLUTION: Can someone help me find the equivalent to the following equation? And do I subtract 29, or just part of it? y = x2 - 8x + 29

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Can someone help me find the equivalent to the following equation? And do I subtract 29, or just part of it? y = x2 - 8x + 29       Log On


   

Question 130392: Can someone help me find the equivalent to the following equation? And do I subtract 29, or just part of it?
y = x2 - 8x + 29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y=1+x%5E2-8+x%2B29 Start with the given equation


y-29=1+x%5E2-8+x Subtract 29 from both sides


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


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


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


y-29=1%28x-4%29%5E2-16 Multiply


y=1%28x-4%29%5E2-16%2B29 Now add 29 to both sides to isolate y


y=1%28x-4%29%5E2%2B13 Combine like terms



Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=4, and k=13. Remember (h,k) is the vertex and "a" is the stretch/compression factor. Also "a" tells us which direction the parabola opens.



So in this case the vertex is (4,13) and the parabola opens upward since a%3E0


Check:

Notice if we graph the original equation y=1x%5E2-8x%2B29 we get:

graph%28500%2C500%2C-20%2C20%2C-20%2C20%2C1x%5E2-8x%2B29%29 Graph of y=1x%5E2-8x%2B29. Notice how the vertex is (4,13).


Notice if we graph the final equation y=1%28x-4%29%5E2%2B13 we get:

graph%28500%2C500%2C-20%2C20%2C-20%2C20%2C1%28x-4%29%5E2%2B13%29 Graph of y=1%28x-4%29%5E2%2B13. Notice how the vertex is also (4,13).


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