﻿ Exponential Growth, its properties, how graph relates to the equation and formula--Visual Lesson | Math Warehouse

# Exponential Growth Equations and Graphs

Formula and graph for exponential growth equations

### Properties

• Property #1) rate of growth starts slow and increases ( Read on, to learn more about this property, which is the primary focus of this web page)
• Property #2) The domain is
all real numbers . $$\{x: x \in \mathbb{R}\}$$
• Property #3) The range is
all positive real numbers greater than 0 .$$\{ y: y > 0 \}$$

• Property #4) It is a one-to-one function
• Property #5) The graph is asymptotic with
• Property #6) The value of 'b' in the general equation must be
greater than 1
• Property #7) The inverse of exponential growth is
logarithmic functions.

Of note: Exponential decay is not the inverse of exponential growth.

### How Equation relates to Graph

In the general example below, '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 ) .

### The Graph

Property #1) Rate of growth of exponential growth increases more and more, until it becomes massive!

As the graph on the left shows, at first, exponential growth does not increase much, but the rate of growth of begins increasing more and more until the rate of increase becomes massive.

### Table of Values

The table of values for the exponential growth equation $$y = 9^x$$ demonstrates the same property-growth rate starts slow and soon gets massive

At first the rate of increase is small, but the pace increases and soon enough the rate of increase is massive.

Just look at the difference between x = 7 and x = 8, the value of the function increases by more than 38 MILLION!

#### 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) = x 3, 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.

### 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

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

#### 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

$$y = 2^x$$

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