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# Exponential Growth Equations and Graphs

Formula and graph for exponential growth equations

### How Equation relates to Graph

Graph 1

In the general example shown in Graph 1, 'a' stands for the initial amount, and 'b' is any real number that is greater than 1.

The graph's asymptote at x -axis

What about when b is exactly equal to 1?

In this case, you are not dealing with an exponential equation, but rather a linear equation. (Link) $y = ab^x \\ y = a(1)^x \\ y = a$

As you can see from the work above, and the graph, when b is 1, you end up with the equation of a horizontal line ( Link ) .

#### How does this compare to other graphs/functions?

As the graph below shows, exponential growth

• at first, has a lower rate of growth than the linear equation f(x) =50x
• at first, has a slower rate of growth than a cubic function like f(x) = x3, but eventually the growth rate of an exponential function f(x) = 2 x , increases more and more -- until the exponential growth function has the greatest value and rate of growth!

Moral of the story: Exponential growth eventually grows at massive rates, even though it starts out growing slowly. This 'trend' is true and, when compared to other graphs, exponential growth eventually outpaces most other functions's rate of increase.

### Role of a in $$y=ab^x$$

For $$y = \color{red}{a} \cdot b^x$$, a determines the y-intercept.

### Some Examples

Three different exponential growth functions are graphed in the diagram below.

The graph on the left helps show the role of 'b' in the $$y = a \cdot b^x$$

Namely, the greater the value of b

• the 'steeper' the curve looks
• the greater the rate of growth

#### What would you do?

Take the Poll! (and think about exponential growth)

(This poll has its own page)
Option 1

You can have $1000 a year for twenty years Option 2 You can get$1 for the first year, $2 for the second,$4 for the 3rd, doubling the amount very year -- for twenty years.

### Closer Look at Graph/Equation

Below is a picture of the most commonly studied example of exponential growth, the equation $$y = 2^x$$