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- Two videos ago, I told you that when you're dealing with
- an exponential function, that when you increase x by a
- constant amount that the y values will increase by a
- constant factor.
- When we increased x from 0 to 1, we increased y
- by a factor of 1.5.
- We multiplied 1.5 times 120 to get 180.
- When we increased x by 1 again, once again, y increased
- by a factor of 1.5.
- We multiplied 1.5 times 180.
- What I want to do in this video is to kind of quickly
- show you just going from the basic definition of an
- exponential function, why this is the case.
- Why this works for exponential functions.
- So let's just start with a general exponential, y is
- equal to A times r to the x.
- And let's look at some x values and then their
- corresponding y values.
- These are my x values, those are my y values over there.
- Now let's say that x is equal to, let me just pick an
- arbitrary number.
- Let's say that x is equal to b.
- I haven't used that variable yet.
- And what's y going to be equal to?
- It's going to be equal to A times r to the bth power.
- Now, let's increase x by 1.
- This would also work if you increased x by really, just a
- constant factor.
- But let's increase it by 1.
- So we're now going to look at x is equal to b plus 1.
- Then y is going to be equal to A times r to
- the b plus 1 power.
- Let's do it one more time.
- Let's increase x by 1.
- You get b plus 2.
- And now Ax-- sorry.
- Now y would be A times r to the b plus 2 power.
- Now let's see what happens when you take the ratio of
- each successive y terms that correspond to the x terms when
- you increase by a constant factor.
- And notice, every time we increased x by exactly 1.
- If this was 0, this was 1, this is 2.
- If this is negative 5, this would be negative 4, this be
- negative 3.
- So let's see what happens with our y.
- So over here, if we look at the factor that y changed by,
- we could just divide Ar to the b plus 1 by Ar to the b and
- what do we get?
- Well you might already recognize you could use
- straight up exponent rules right here.
- The A's cancel out and then this is going to be equal to r
- to the b plus 1 minus b.
- Same base.
- You're taking the quotient.
- You subtract exponents.
- So this is just going to be equal to r.
- Let's see what happens when we go from here to here.
- Once again, take the y value when x is equal to b plus 2,
- so Ar to the b plus 2.
- Divide it by the previous y value.
- Divide it by Ar to the b plus 1.
- These cancel out and we're left with r to the b plus 2
- minus b plus 1, which is equal to r to the b plus 2
- minus b minus 1.
- These cancel out.
- 2 minus 1.
- Once again, you're just left with r.
- So hopefully that satisfies any curiosity you might have
- had of why you're able to use that as the way to evaluate
- whether this data was exponential.
- Why we're able to just say, hey, let's take this term
- divided by the previous term and see what number we get.
- This term divided by that term and if we get the same number
- every time, we're dealing with an exponential function.