Simple minimum/maximum question

[DeanBH]

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

 

[HallsofIvy]

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 presume you mean (x+5)²- 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

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.

 

[DeanBH]

Can anyone tell me how i find the line of symmetry of this curve, and why it is that 8D

 

[HallsofIvy]

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