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Homework Help: Simple minimum/maximum question

  1. Apr 24, 2008 #1
    I did a past paper question whereby I carried the answer (X+5)-6 through.

    it then asked for the minimum points of the graph y=x^2 + 10x + 19. which is what i made into (X+5)-6.

    I know i have to take the +5 and change its sign to -. and that's the minimum of X. and the -6 without change is the minimum of Y.


    I just wondered, can anyone explain to me why this is so?


    Furthermore, what would I do if it asked for a maximum. I would show an attempt, but i don't even understand why this is a minimum so it's hard for me to find out the maximum.
    ( this isn't actually a question I'm just interested.)


    thanks :P
     
  2. jcsd
  3. Apr 24, 2008 #2

    HallsofIvy

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    I presume you mean (x+5)2- 6.

    A square is never negative. If y= (x- 5)2- 6, It is always "-6 plus something". If x- 5= 0, which is the same as x= 5 (adding 5 to both sides. I cringe when I read somenthing like "take the+ 5 and change its sign to -"!!), y= 0- 6= -6. For any other x, x- 5 is non-zero, (x- 6)2 is positive and (x- 5)2- 6 is larger that 6.

    If the problem asked for a maximum, there is something wrong with the problem! The graph of y= (x- 5)2- 6 is a parabola that opens upward: its "vertex" is at the lowest point (5, -6). There is no highest point.

    However, if the problem were y= -(x- 5)2- 6 (that's the same as y= x2+ 10 x+ 31), then you can argue that when x= 5, y= -02+ 31= 31 but for any other value of x, y= -(a positive number)+ 31 and so is less than 31. In this case, the graph is a parabola that opens downward. (5, 31) is the highest point on the parabola and 31 is the maximum value of y.
     
  4. Apr 24, 2008 #3
    Can anyone tell me how i find the line of symmetry of this curve, and why it is that 8D
     
  5. Apr 25, 2008 #4

    HallsofIvy

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    Do you know the definition of "line of symmetry"?
     
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