HELP! The following function shows the relationship between the selling prices, and profit P(s), in dollars, for a company:
P(s) = -20s2 + 1,400s - 12,000
Which statement best describes the intervals where the company's profit increases, decreases, or records a maximum?
a) It is least when the selling price is $30.
b) It is greatest when the selling price is $30.
c) It decreases when the selling price increases from $10 to $35.
d) It increases when the selling price increases from $10 to $35.
Explanation for answer would be greatly appreciated. Thanks.

- anonymous

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- Anaise

hi

- SolomonZelman

Calc 1?

- SolomonZelman

Or just a vertex problem?

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## More answers

- anonymous

I wish...but nah Algebra I

- SolomonZelman

Well, we don't need calculus really, since it is a parabola..

- SolomonZelman

\(\color{#000000 }{ \displaystyle p(s)=-20s^2+1400s-12000 }\)

- SolomonZelman

Can you factor the left side out of -20?

- anonymous

Sure, one sec.

- anonymous

-20(s^2-700s+600)

- SolomonZelman

wait the middle

- SolomonZelman

middle term is wrong.

- anonymous

Sorry, -20 (s^2 + 70s + 600)

- anonymous

Wait isn't that wrong?

- anonymous

Oh man i factored 14000 instead, one sec

- SolomonZelman

\(\color{#000000 }{ \displaystyle -20 (s^2 - 70s + 600) }\)

- anonymous

-20 (s-10) (s-60)

- SolomonZelman

no need for complete factorization.

- SolomonZelman

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\)
is what you need at this point.

- anonymous

Okay i got that

- SolomonZelman

you need to complete the square...
What would you add
\(\color{#000000 }{ \displaystyle s^2 - 70s + ~? }\)
to make a perfect square trinomial?

- SolomonZelman

Note that,
\(\color{#000000 }{ \displaystyle x^2+2ax+a^2\color{grey}{~~~~~=(x+a)^2}}\)
is something that you want to obtain

- SolomonZelman

so the 2a corresponds (in our case), to -70, right?

- anonymous

It's not a perfect square, right?

- SolomonZelman

the x²+2ax+a² is a perf. sq.

- SolomonZelman

your polynomial inside the parenthesis is not,,,

- anonymous

Yea, in the parenthesis, the first term and last term are perfect squares, but the middle one isn't, right?

- SolomonZelman

but if you tell me the missing number for
\(\color{#000000 }{ \displaystyle s^2 - 70s +~{\rm what?} }\)
then I can show you a trick.

- SolomonZelman

"what?" will make the s²-70s a perfect square trinomial when added?

- anonymous

I don't know

- SolomonZelman

\(s^2+2as+a^2\)
is the form you want

- SolomonZelman

you have
\(s^2-70s\)

- SolomonZelman

So -70s, corresponds to the "2as" peace, right?

- SolomonZelman

So, what is the "a" in our case?

- anonymous

-35?

- SolomonZelman

yup

- SolomonZelman

and "a²" is what?

- anonymous

YAY!

- SolomonZelman

ok, next question i asked...

- anonymous

(-35)^2

- SolomonZelman

yes, and that would be?

- anonymous

1225

- SolomonZelman

yup

- SolomonZelman

So, I will show you the trick now...

- SolomonZelman

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600) }\)
You already have 600 in parenthesis, and you need to make that a 1225. (For this to be a perfect square trinomial)
\(\color{#000000 }{ \displaystyle 600+x=1225 }\)
\(\color{#000000 }{ \displaystyle x=625 }\)
But, you can't just add 625, you will have to use something that I refer to as the "magic zero", and you will see what I am talking about NOW...
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600+625-625) }\)
Now, we would like to get rid of -625 in parenthesis, but we can't just erase it. We will take it out, by multiplying times -20.

- SolomonZelman

\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 600+625) +(-20)(-625)}\)
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 70s + 1225) +1250}\)
and recall the form, \(s^2+2as+a^2=(s+a)^2\)
We have already clarified that:
a=-35
a²=1225
2a=-70
\(\color{#000000 }{ \displaystyle p(s)=-20 (s^2 - 2(35)s +(-35)^2) +1250}\)

- SolomonZelman

And we apply the form now!
\(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\)

- SolomonZelman

when you digest this info, let me know...

- anonymous

One moment, still digesting

- anonymous

I got it. What's next?

- SolomonZelman

\(\color{#000000 }{ \displaystyle p(s)=-20 (s - 35)^2 +1250}\)
can you tell me if this parabola opens up or down (and why)?

- anonymous

down? i think..

- SolomonZelman

yes, and why?

- anonymous

negative coefficient?

- SolomonZelman

yes, the leading coefficient is negative...
fabulous!

- SolomonZelman

Since the parabola opens down, the vertex is the maximum point of the parabola. (The other points are all lower)

- SolomonZelman

So you need to find the vertex, and for this there is a rule...

- SolomonZelman

\(\color{#000000 }{ \displaystyle y(x)=a(x-k)^2+h }\)
will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)

- anonymous

(1250,-35)?

- anonymous

Sorry (1250, 35)

- SolomonZelman

not k,h
the other way around

- SolomonZelman

my fault

- SolomonZelman

\(\color{#000000 }{ \displaystyle y(x)=a(x-h)^2+k }\)
will have a vertex of \(\color{#000000 }{ \displaystyle (h,k) }\)

- SolomonZelman

that is the form

- anonymous

(35, 1250)

- SolomonZelman

yes, correct, and I apologize for my mistake.

- anonymous

No worries, you have been amazing help!

- anonymous

And guidance

- SolomonZelman

And the vertex of (35,1225) indicates that the maximum profit is 1225, and it occurs when you sell 35 dollars per item.

- SolomonZelman

Which statement best describes the intervals where the company's profit increases, decreases, or records a maximum?
a) It is least when the selling price is $30.
b) It is greatest when the selling price is $30.
c) It decreases when the selling price increases from $10 to $35.
d) It increases when the selling price increases from $10 to $35.

- anonymous

d?

- SolomonZelman

you are close.

- anonymous

c?

- SolomonZelman

can you sketch (somehow) a parabola that opens down please?

- SolomonZelman

any picture that looks like opening-down parabola would suffice.

- anonymous

|dw:1450410288830:dw|

- SolomonZelman

good

- SolomonZelman

|dw:1450410304542:dw|

- SolomonZelman

|dw:1450410339366:dw|

- SolomonZelman

is the slope of this line positive or negative?

- anonymous

positive

- SolomonZelman

good

- SolomonZelman

|dw:1450410402632:dw|

- SolomonZelman

is the slope of this (red) tangent line positive or negative?

- anonymous

positive

- SolomonZelman

ok

- SolomonZelman

|dw:1450410507209:dw|

- SolomonZelman

all of these Lines (A, B, C and L) are tangent to the parabola

- SolomonZelman

Which lines have negative slopes and which have positive slopes?

- anonymous

A and C have positive, B and L have negative

- SolomonZelman

awesome!

- SolomonZelman

negative slope indicates that the price is increasing, right?

- SolomonZelman

yes or no?

- anonymous

No..?

- SolomonZelman

Yes,

- SolomonZelman

the price is decreasing if slope is negative

- SolomonZelman

And is increasing if the slope is positive

- SolomonZelman

|dw:1450410804794:dw|

- SolomonZelman

So does the price increase or decrease from s=10 and up to s=35?

- anonymous

increase

- SolomonZelman

yes, and thus you know the answer...

- anonymous

Wait is it c or d?

- anonymous

d, right?

- SolomonZelman

it increases up to 35....

- SolomonZelman

D

- anonymous

Gotcha, thanks!!

- SolomonZelman

yw

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