Question

an electronics manufacturer recently created a new version of a popular device. it also created this function to represent the profit, P(x), in tens of thousands of dollars, that the company will earn based on manufacturing x thousand devices: P(x)=-0.16x^2+21.6x-400. The profit functions for the first version of the device was similar to the profit function for the new version. As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version. for the value x=40, the original profit function is half the size of the new profit functi-

Answer 1

-ion. write two function transformations in terms of P(x) that could represent the original profit function?

Answer 2

@mathmate

Answer 3

@jewelliasmith1
Have you finished reading the question?

Answer 4

yes @mathmate

Answer 5

Good!
This problem is expressed in real-life application form, which makes it look like it is long. The mathematical part can be resumed into much simpler terms.
Would you like to explain to me the mathematical aspect of the question?

Answer 6

im so confused

Answer 7

Can you at least extract what is given to you from the question? Whatever comes to your mind?
This practice will help you solve other word problems later on.

Answer 8

p(40)=-0.16(40)^2+1.6(40)-400

Answer 9

That's good!
What does this function represent?

Answer 10

the device thingy

Answer 11

By the way, the function is
p(40)=-0.16(40)^2+\(\color{red}{2}\)1.6(40)-400

Answer 12

It is the amount of profit when 40 devices are sold.
Reread the question and check if what I said is true. I make mistakes too! lol

Answer 13

yea

Answer 14

Good!
Now, recall that P(x) represents the profit for selling x units with the \(new\) device.
The old device has a profit function (I will call it PT(x)) which is a transformation of P(x), right?

Answer 15

i guess

Answer 16

Reread the question to make sure you understand the same way as I said earlier. Otherwise you'd be following me blindly, and you won't be able to solve your next problem!

Answer 17

ok

Answer 18

so what do i put as the answer

Answer 19

You'll need to do more before you get the answer!
If that's understood, so far, you need to decide on a transformation of P(x) to PT(x).
Two transformations are possible,
1. you can do a horizontal stretch, i.e. PT(x)=P(kx) with the stretch factor k, or
2. you can do a horizontal translation, i.e. PT(x)=P(x-b)
where k and b are constants to be determined. We don't know what they are, yet.

Answer 20

im so confused

Answer 21

That is why you need to understand the question as a start.
Please reread the question, especially the part:

The profit function for the first version of the device was similar to the profit function for the new version. As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version.

Answer 22

Is it a little less confusing now?
The question explains that the profit function for the old product is a transformation of that of the new one, hence
PT(x)=transformation of P(x)
And we can try the transformation vertical stretch, or horizontal translation, both are possible.

Answer 23

Perhaps reading the following will help to clarify a little:

http://www.purplemath.com/modules/fcntrans.htm

Answer 24

oh ur talking about graphs

Answer 25

Yes, that too!
Actually, the graph of P(x) is as follows:
http://prntscr.com/bg9ow2

Answer 26

ok

Answer 27

To proceed, we will now make use of the information:

For the value x=40, the original profit function is half the size of the new profit functiion.

which can be expressed in a mathematical equation:
PT(40)=(1/2) P(40).

Does that make sense to you?

Answer 28

yea

Answer 29

Good, so let's start with the vertical stretch as a transformation.
A vertical stretch is simply PT(x)=a P(x), where a is a constant.
Putting in
PT(40)=aP(40)=(1/2)P(40), can you solve for a?
hint: use the second part: aP(40)=(1/2)P(40) to solve for a.

Answer 30

i dont get it

Answer 31

The question tells us:
"As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version. "

So we're making a transformation (of any kind, but we chose the simplest kind, a vertical stretch) to satisfy the other requirement:
" For the value x=40, the original profit function is half the size of the new profit functiion."

Therefore we come up with the previous equation:
PT(40)=aP(40)=(1/2)P(40), can you solve for a?
hint: use the second part: aP(40)=(1/2)P(40) to solve for a.

Answer 32

ok

Answer 33

So now you need to solve for "a" in the equation in order to complete the definition of the transformation.

Answer 34

ok

Answer 35

5.2

Answer 36

Not quite.
The equation to solve is

\(\large a~ P(40)=(1/2)~P(40)\)
we need the value of "a".
Try to transpose or cancel common factors.

Answer 37

Perhaps this might help you!

https://www.khanacademy.org/math/k-8-grades/cc-eighth-grade-math/cc-8th-solving-equations