Precaution when scaling down (dividing) Quadratic Function

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Suwandy

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Hi guys, i'm here asking weirdness that i found during learning Quadratic Function. I'm still not used with posting in forum, so i don't know how to write equation here. Sorry.
I got this question in KhanAcademy exercises:

y= -4x2 -32x -68. The goal is to find vertex coordinate

The usual method to do this was averaging both x-value from the solutions (we can get them by factoring, formula, or completing the square). The other method is transforming this standard form into vertex form (i only know these 2 methods). The idea was every method should be yielding same result (that's my belief) so i'm trying out all possible methods.

Using method 2 (turn into vertex form):
I noticed that a, b, c coefficients were asking to be divided by -4 (the temptation was great!) so here are my steps:
  1. y= x2 +8x +17
  2. Completing the square: y= (x2 +8x +16) -16 +17, which turn into y= (x + 4)2 + 1
  3. So i conclude the vertex coordinate was (-4, 1)

Checking the key answer, i was wrong!! the correct answer was (-4, -4). At this point, i retried using other method (finding solutions). Finding solutions mean finding x-value when y=0. I started again by dividing the function with -4:
  1. 0= x2 +8x +17
  2. Completing the square: 0= (x2 +8x +16) -16 +17, which turn into -1= (x + 4)2
  3. Umm, this is bad. Continuing this i will have sqrt(-1) = sqrt((x + 4)2). Which is not real number so i messed up here.

Now, i tried using the sacred formula! :
  1. 0= x2 +8x +17 (a=1, b=8, c=17)
  2. (-8 +- sqrt(64 - 4*1*17)) / 2*1
  3. whoops this also won't make it! i also get imaginary number here which also messed everything up.

Tracing back what i did wrong, i thought dividing by -4 is the real culprit. Using vertex form again, i tried without dividing by -4:


  1. y= -4x2 -32x -68
  2. y= -4(x2 +8x +17)
  3. y= -4((x2 +8x +16) -16 +17)
  4. y= -4((x+4)2 +1)
  5. y= -4(x+4)2 -4
  6. Yay! i finally get the vertex (-4, -4)

So the big question: Why dividing the function by -4 screw everything up??

my hypothesis is dividing by -4 screw the y function altogether:
  1. y= -4x2 -32x -68
  2. divide by-4: (y/-4) = x2 +8x +17
is this true? if yes, when can we safely scale down a function by dividing them??
 
Last edited:
Suwandy said:
Hi guys, i'm here asking weirdness that i found during learning Quadratic Function. I'm still not used with posting in forum, so i don't know how to write equation here. Sorry.
I got this question in KhanAcademy exercises:

y= -4x2 -32x -68. The goal is to find vertex coordinate

The usual method to do this was averaging both x-value from the solutions (we can get them by factoring, formula, or completing the square). The other method is transforming this standard form into vertex form (i only know these 2 methods). The idea was every method should be yielding same result (that's my belief) so i'm trying out all possible methods.

Using method 2 (turn into vertex form):
I noticed that a, b, c coefficients were asking to be divided by -4 (the temptation was great!) so here are my steps:
  1. y= x2 +8x +17
  2. Completing the square: y= (x2 +8x +16) -16 +17, which turn into y= (x + 4)2 + 1
  3. So i conclude the vertex coordinate was (-4, 1)

Checking the key answer, i was wrong!! the correct answer was (-4, -4). At this point, i retried using other method (finding solutions). Finding solutions mean finding x-value when y=0. I started again by dividing the function with -4:
  1. 0= x2 +8x +17
  2. Completing the square: 0= (x2 +8x +16) -16 +17, which turn into -1= (x + 4)2
  3. Umm, this is bad. Continuing this i will have sqrt(-1) = sqrt((x + 4)2). Which is not real number so i messed up here.

Now, i tried using the sacred formula! :
  1. 0= x2 +8x +17 (a=1, b=8, c=17)
  2. (-8 +- sqrt(64 - 4*1*17)) / 2*1
  3. whoops this also won't make it! i also get imaginary number here which also messed everything up.

Tracing back what i did wrong, i thought dividing by -4 is the real culprit. Using vertex form again, i tried without dividing by -4:


  1. y= -4x2 -32x -68
  2. y= -4(x2 +8x +17)
  3. y= -4((x2 +8x +16) -16 +17)
  4. y= -4((x+4)2 +1)
  5. y= -4(x+4)2 -4
  6. Yay! i finally get the vertex (-4, -4)

So the big question: Why dividing the function by -4 screw everything up??

my hypothesis is dividing by -4 screw the y function altogether:
  1. y= -4x2 -32x -68
  2. divide by-4: (y/-4) = x2 +8x +17
is this true? if yes, when can we safely scale down a function by dividing them??
Click to expand...
The problem is in how it is being applied. All of your initial work is correct except for one small step but first lets change the problem just a little bit so that the other methods work
y= -4x2 -32x - 60.
Now for the second and third method dividing through by -4 we get
0 = x2 + 8 x + 15 = (x2 + 8 x + 16) + 15 - 16 = (x+4)2 - 1
So the values are -4\(\displaystyle \pm\)1 or -5 and -3. The average is -4, so -4 is the x part of the vertex.
However, and this is where the mistake was made, y is not x2 + 8 x + 15 (= (x+4)2 - 1), it is given by
y = -4 (x2 + 8 x + 15) = -4 [(x+4)2 - 1] = -4 (x+4)2 + 4
or in the case of the actual problem
y = -4 (x+4)2 - 4

The reason the methods worked to find the x value is essentially that completing the square or using the quadratic formula can be done in either the factored form (factor the -4 out of the equation) or non-factored form to obtain the x value. That is 0/(-4) is still 0. However, when you write the equation for y, you have to remember to put back in the factor(s) you took out along the way. That is y/(-4) is not y.

Oh, in case you just slapped yourself up side the head, be careful, that can hurt. I know from personal experience.
 
Ishuda said:
The reason the methods worked to find the x value is essentially that completing the square or using the quadratic formula can be done in either the factored form (factor the -4 out of the equation) or non-factored form to obtain the x value. That is 0/(-4) is still 0. However, when you write the equation for y, you have to remember to put back in the factor(s) you took out along the way. That is y/(-4) is not y.
Click to expand...
B-but, i tried quadratic formula on both factored & non-factored form of y= -4x2 -32x -68. Both gave me imaginary number...

ok, so (y/-4) is they key in this problem, got it!
Thanks Ishuda!

Ishuda said:
Oh, in case you just slapped yourself up side the head, be careful, that can hurt. I know from personal experience.
Click to expand...

LoL, i've been doing this since high school! no wonder i'm getting dumber! hahahaha
 
BTW: On that finding the zeros method: If x1 and x2 are the zeros of y, then the vertex occurs at x=(x1+x2)/2. Lets do the quadratic solution method to find out zeros for the general equation
a x2 + b x + c = 0
To simplify matters let
d2 = b2 - 4ac
and d be 'the positive square root'. We then have
\(\displaystyle x_1\, =\, \frac{-b\, +\, d}{2\, a}\)
and
\(\displaystyle x_2\, =\, \frac{-b\, -\, d}{2\, a}\)
Regardless* of the value of d, what is (x1+x2)/2



*I would have used irregardless here since I like the word and it was officially entered into the Websters(?) dictionary more than 50 years ago. However the 'word police' are lurking and disapprove of the use. So, in order to keep them from suffering an apoplectic fit, I don't use the word.
 
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