Freshener Air Problem.

Question

A company that makes air fresheners has determined with x milliliters of liquid, their product lasts for Q(x)days. Based on their current formula, a 20 milliliter (mL) container will last for 21 days. Thus we know Q(20)=21.

The research and development team have come up with some new formulas. For each case, determine where the point (20,21) would move to on a graph of the transformed function, and write a new formula in terms of transformations (example: y=2Q(x))

A) Formula A uses 5 fewer mL of liquid and lasts 3 days longer. 

B) Formula B uses 1/2 as much liquid, and lasts for just as long as the original. 

C) Formula C uses twice as much liquid, and lasts three times as long as the original.

Answer 1

Lets do this as if we didn't remember the transformation formulas/rules, or are a bit unclear on how to apply them, or maybe we don't even know them.

A) Now we only know that Q(20)= 21 But lets think of it in terms of (x,y) where x is the amount of liquid, y is the number of days the air freshener lasts

Now we want want to use 5ml fewer liquid. Lets think: we only have Q(20) = 21 as a point, or (20,21), so lets use that to figure this out.

instead of 20 we only want to use x= 15ml. But we don't know what Q(15) is, only Q(20). So what do we do to 15 to get to 20?

We add 5. so we need x+5 to be our input. Instead of Q(x) we are going to need Q(x+5)

How long we want the air freshener to last? 3 days longer. How do we express that? So far we have

y=Q(x+5) and if we input 15 as our x, we would only get y=Q(20) = 21 days only. So we have to adjust the formula to get 3 more days, so we just add 3:

y=Q(x+5) + 3

Let's see if it make sense? So if we put x=15 ml of liquid, we get

y = Q(15+5) +3 --›y=Q(20)+3 -->y=21+3 -->y=24. So we used 5 less ml liquid to get an air freshener that lasts 3 days longer

Finally, our point (20,21) changed to (15,24)

B) Half as much liquid - so lets use (20,21) as the only example we have to see if we can think about this correctly. This would mean we would be able to use only 10ml of liquid for the freshener to last the same 21 days. So we have x=10, but since Q(20)=21 and not Q(10) we need to double our x, or

y=Q(2x)

Mental check: If I used x=10 ml of liquid,

y=Q(2*10)

y=Q(20)=21

Our point changed from (20,21) to (10,21)

Note that since we wanted the same number of days, we did not have to add or do anything to Q(2x)

C) Lets see if we get the hang of this. Start with (20,21) lets see what happens

Twice as much liquid - so now we need x = 40 ml ---›In order to get back to Q(20), the value we know, what do we have to do to x? Take half of it

Q(x/2)

Last 3 times as long. So instead of 21 days, I need 21*3 = 63 days. So I need to multiply what Q(x/2) by 3:

y = 3Q(x/2)

Check if I put in x=40, I get

y=3Q(40/2)

y=3Q(20)

y=3(21)=63 so yes, we used twice the amount of liquid but it lasted 3 days longer

Our original point (20,21) would move to (40,63)

Hope this helps!