Making an exponential function given points..

Question

johnweldon1993 · Answer

@briannamooney

johnweldon1993 · Answer

So remember the general formula for an exponential looks like

$$\large y = ab^x$$

johnweldon1993 · Answer

So what are 2 points you have again?

anonymous · Answer

(2,1210)

johnweldon1993 · Answer

is that 1 point?

(2 , 1210)  ?

I would need another as well

anonymous · Answer

(1,1100)

johnweldon1993 · Answer

Alright good...so we have 2 points

(1 , 1100) and (2 , 1210)

So we take our formula and plug in point 1

$$\large y = ab^x$$
becomes
$$\large 1100 = ab^1$$

and we do the same with the 2nd point

$$\large y = ab^x$$
becomes
$$\large 1210 = ab^2$$

johnweldon1993 · Answer

Now what we do...is divide these 2 functions

$$\large \frac{1210 = ab^2}{1100 = ab}$$

Notice

$$\large \frac{1210 = \cancel{a}b\cancel{^2}}{1100 = \cancel{ab}}$$

what cancels out

What we have is

$$\large b= \frac{1210}{1100}$$

anonymous · Answer

so far so good

johnweldon1993 · Answer

And once we find out what that equals *pgpilot posted the answer above* we would plug back in that 'b' and find out what 'a' equals

so

$$\large y = ab^x$$

we'll use the first point again

$$\large 1100 = a(\frac{11}{10})^1$$
or just

$$\large 1100 = a(\frac{11}{10})$$

Now we just solve for 'a'

anonymous · Answer

how did you get 11/10 , by dividing 1200/1100?

johnweldon1993 · Answer

$$\large \frac{1210}{1100}$$

If you divide both numbers by 110 we get

$$\large \frac{11}{10}$$

anonymous · Answer

oh okay gotchya

johnweldon1993 · Answer

So yes...now that we found what 'a' and 'b' equal we have our function of your exponential

$$\large y = 1000(\frac{11}{10})^x$$

everything make sense? :)

anonymous · Answer

did A change to Y ?

johnweldon1993 · Answer

Nope...

so the 2 points we had were

$$\large (1,1100) (2,1210)$$

a we found to be 1000
and
b we found to be 11/10

anonymous · Answer

Now it makes sence , thank you :)

johnweldon1993 · Answer

Of course :)

anonymous · Answer

So if we plugged in x=3 it should equal (3,1331) ? Cause thats the next points

johnweldon1993 · Answer

Lets try it out :)

$$\large y = 1000(\frac{11}{10})^x$$

would become

$$\large 1331 = 1000(\frac{11}{10})^3$$
$$\large 1331 = 1000(\frac{11^3}{10^3})$$
Now what does 10^3 equal? 10 x 10 x 10 = 1000 right? so we have
$$\large 1331 = 1000(\frac{11^3}{1000})$$

Notice

$$\large 1331 = \cancel{1000}(\frac{11^3}{\cancel{1000}})$$

So we have

$$\large 1331 = 11^3$$

is that correct?

11 x 11 = 121\]

121 x 11 = 1331 $\large \color \red{\checkmark}$

anonymous · Answer

that was confusing lol , i just typed in a calculator 11/10^3 , i didnt know you had to square both number

johnweldon1993 · Answer

Ohhh no no no you have to do it to both lol

$$\large (\frac{11}{10})^3 = \frac{11^3}{10^3}$$

anonymous · Answer

oh okay now i know

johnweldon1993 · Answer

Good :) so, does everything make sense now? :)

anonymous · Answer

yes ! thank you so much

johnweldon1993 · Answer

Anytime :) I'm here if you need anything else :D

anonymous · Answer

why would this be a exponential function ?

johnweldon1993 · Answer

Well, because it certainly isn't linear

We go up from x = 1 to 2 and then to 3....that is linear...

but for the y-values...we have

1100, 1210,1331

Is that linear? does it increase by the same amount every time?

well we go from 1100 to 1210...that is an increase of 110

and then we go from 1210 to 1331....that is an increase of 121

Doesn't look linear to me :)

anonymous · Answer

Yes your right :)