Question

If f(x) varies directly with x, and f(x) = 12 when x = 8, write the direct linear variation equation.

Answer 1

***ALWAYS REMEMBER y = f(x)***

When they say that f(x) varies directly with x, it means that,
As x increases, f(x) also increases, and as x decreases f(x) decreases.

Take for example: f(x) = x



So in your example when they say that

f(x) varies directly with x we say

f(x) \(\alpha\) x
Or
f(x) = \(k.x\) [Where k is a constant of proportionality.

So in your question it is given that when,


f(8) = 12 or
f(8) = k.8 [Because here, x = 8]

Now if you find the value of 'k', then only can you write the equation, as y = k.x

Answer 2

wow i do not get any of that

Answer 3

Here's the deal... first, let's end confusion and let y = f(x)
means replace f(x) with y.
y looks a lot less menacing than f(x), don't you agree?
We'll switch back later.

Okay, ready?

Answer 4

yes!

Answer 5

When two values are in direct variation, it means their QUOTIENT is constant.
So, since y and x are in direct variation, it means
\[\Large \frac y x = k\]

for some constant k, correct?

Answer 6

okay..

Answer 7

I don't like the uncertainty.
If there's something you don't get, tell me :D

Answer 8

no, no i get it so far

Answer 9

Okay... what you must understand is that k is a CONSTANT.
It doesn't change.
All we have to do now is find out exactly what k is.
Any idea how?

Answer 10

no..

Answer 11

Of course not (not meant as an insult ^_^)
That's why you're here :D

So, we know that
\[\Large \frac yx = k\] and k is a constant that doesn't change... no matter what x and y are.

It just so happens we were given specific values for x and y, do you know them?
(mentioned in the question)

Answer 12

y = 12, x = 8

Answer 13

that's right.
So we can plug those in, and we get (I wouldn't mind a drum roll)
\[\Large \frac{12}{8}= k\]
And suddenly, we have an exact value for k

So, simplify it, what is k equal to?

Answer 14

3/2?

Answer 15

Good. So now, we know k = 3/2

And we know k doesn't change.
So we have
\[\Large \frac yx = \frac32\]
Separate y.
Make y stand alone on the left-side.

Answer 16

y = 3/2x?

Answer 17

\[\Large y = \frac32x\]
That's right.
Now you can replace y with f(x)
and get your final answer ^_^

Answer 18

wow thank you so much!

Answer 19

Your final answer, post it, before thanking me XD

Answer 20

f(x) = 3/2x (:

Answer 21

And actually, @AkashdeepDeb has said pretty much the same things, only he said them all at once, which may have been a bit overwhelming ^_^
Kudos to him, nonetheless

Answer 22

yeah thank you both, it was a bit confusing at first

Answer 23

Now, remember these, these may be helpful in the future...
If x and y are in DIRECT variation, it means their QUOTIENT is constant
\[\Large \frac y x = k\]

If x and y are in INVERSE variation, it means their PRODUCT is constant
\[\large xy = k\]

Get it?
Got it?
Good.

:D

Answer 24

yes thank you