Exponential Growth

In this video, we are going to look at exponential growth.

$y=a \times (b)^x$
For a graph to demonstrate exponential growth, the coefficient, a, must be positive, and the base of the exponent, b, must be greater than 1. The graph will follow a path that curves upward to the right. These graphs grow by a factor, as opposed to a constant value.
For example:
The graph of
$y=2 \times (1.2)^x$
curves upward after plotting points as shown in the video, and demonstrates exponential growth. This graph grows by a factor of 1.2.
When comparing a linear graph to an exponential growth graph, as more time goes on, the exponential growth tends to grow by larger quantities than the linear growth.

Video-Lesson Transcript

Let’s go over exponential growth.

The basic exponential function is

$y = a \times b^x$ $y = ab^x$

In exponential growth, $a$ is always positive and $b \textgreater 1$.

If we look at the exponential growth graph, it will look like this.

Where each point is some fact greater than the next one.

For example:

$b = 1.2$
$a = 2$

then $y = 2 (1.2)^x$

Let’s see what this table looks like.

Let’s get a financial calculator to solve this.

So when $x = 0$, $y = 2$,
when $x = 1$, $y = 2.4$,
and when $x = 2$, $y = 2.88$.

So the question is how do we get to these points?

Each time we’re just multiplying by $b = 1.2$.

So $2 \times 1.2 = 2.4$, then $2.4 \times 1.2 = 2.88$, and then $2.88 \times 1.2 = 3.456$.

The next point is $x = 3$.

Each time we multiply by $b = 1.2$. It multiplies by the constant $b$.

It’s not going up by a said amount. Here, the increase in $y$ value is increasing in itself.

Whereas when we have a linear graph, it goes up at a constant rate each time.

Here, the jump in $y$ values are linear. The increase or decrease is the same in the entire time.

Just to recap, exponential growth has $x$ as an exponent for it to be exponential, growth is a positive number greater than $1$.

Very important:

Exponential growth grows by a factor.

So you keep on multiplying by a certain number to get a new value.

An example is a compound interest. Each month money may grow by $8\%$ or some value compared to the previous one.

Whereas, linear growth grows by a constant value.

So instead of growing by a certain percentage, it will always grow by a certain amount.

For example, your money grows by $\20$ each month.

On the other hand, exponential growth may grow your money depending on the percentage.

Let’s look at an example:

Two Savings Accounts
$\200$ initially

a) $5\%$ annually
b) $\20$ annually

Let’s start with savings account b where money grows by $\20$ annually.

Year $0$ or initally, we have $\200$.

Then year $1$, we have $220$,
year $2$, we have $240$,
on year $3$, we have $260$,
year $4$, we have $280$,
then year $5$, we have $300$,
year $6$, we have $320$,
year $7$, we have $340$,
and year $8$, we have $360$.

Now, let’s do savings account a where it grows $5\%$ annually.

Year $0$ or initally, we have $\200$.

Then year $1$, we have $210$,
year $2$, we have $220.50$,
year $3$, we have $231.53$,
and on year $4$, we have $243.11$.

Take note that the previous amount grows by $5\%$. So we add the increase to the previous amount to get the new value.

Savings account b still looks further ahead.

Let’s make an equation for this.

$y = 200 (1.05)^x$

Let’s compare it to the one that grows $\20$ every year.

So far, we have no increase of more than $\20$.

From year $10$ to $11$, we have a $\15$ growth. But still not there yet.

After $14$ years, we have an $\18$ growth.

At year $16$, the difference is $\21$.

From year $21$ to $22$, it grew by almost $\30$.

From year $33$ to $34$, it grew by $\50$.

It grows slower, depending on the numbers you’re comparing it to. But after a said amount of time, it will grow at a much greater rate. And will eventually pass the other one.

What if we have a $8\%$ increase?

So right off the bat, it improved by $\16$.

Then at year $4$, it grew by more than $\20$.

From year $9$ to $10$, it improved by $\32$.

The compound interest or exponential growth will quickly pass the linear growth.