Growth and Decay

A factor becomes exponential when it rapidly increases or decreases at the same rate . In Algebra, students often confuse exponential Equations with Quadratic Equations. In a quadratic equation, the power is given but its base is not defined:

The equation is quadratic because the base 'x' is not defined but its power is defined as '2'$\mathrm{y}={\mathrm{x}}^2$ .

However, in exponential Equations, the base is given but the power is not defined:

The equation $\mathrm{y}=2^{\mathrm{x}}$ is exponential because the base is given as '2' while its power 'x' is undefined .

The general algebraic equation of an exponential function is: $y=a{b}^x$

Where a is not equal to 0, b is the base and a positive real Number not equal to 1, and x is the exponent.

Exponential growth and exponential decay are two types of exponential Functions.

What is exponential growth and decay?

Identifying exponential growth and exponential decay?

Exponential growth and decay can be distinguished from each other or identified mathematically.

$\mathrm{y}={\mathrm{ab}}^{\mathrm{x}}$

When a is a positive value and 'b' is the base greater than 1, then it is exponential growth. For example:

$2^{\mathrm{x}}$ note that b = 2, and 2> 1

or

$4\left(3^{\mathrm{x}}\right)$ note that b = 3, and 3> 1

Meanwhile, when a is positive and 'b' is less than 1, then it is exponential decay. For example:

$0.2^{\mathrm{x}}$ note that b = 0.2, and 0.2 <1

or

$5(0.3^{\mathrm{x}})$ note that b = 0.3, and 0.3 <1

Note that in cases where a is negative, it is neither an exponential growth nor decay.

Examples of Exponential Equations

Exponential growth and decay graphs

Introducing Graphs into exponential growth and decay shows what growth or decay looks like. In both cases, you choose a range of values, for example, from -4 to 4. These values will be plotted on the x-axis; the respective y values will be calculated by using the exponential equation.

Also, do not forget that the b value in the exponential equation determines if it is a growth or decay. For b > 1, it is a growth; and for b <1, it is a decay. To help you remember, the b value is in bold in all equations.

Examples of exponential growth and decay graphs

Example 1:

Graph the exponential equation $y=3^x$

Solutions:

Remember to choose a range of values for your x coordinates like -4 to 4 in this case.

So, we need to solve for the value of y. Remember that $y=3^x$

When x = -4

$y=3^{-4}$

$y=\frac{1}{3^4}$ (note that indices, an exponent with a negative sign gives an inverse of the expression $3^4$ )

y = 1/81

Repeat this step for the values x = -3, -2, -1, 0, 1, 2, 3 and 4 and your answer would be 1/27, 1/9, 1/3, 1, 3, 9 , 27 and 81.

Now you can complete your table:

You can see that the y values increase from the left to the right of the table. This means that there has been growth . Go ahead and plot the graph below:

A graph of exponential growth for b> 1

From the exponential growth graph, the following can be observed:

The graph increases steeply as the value of x becomes positive.

There is no x-intercept as the curve fails to intersect with the x-axis even as y values come close to the x-axis.

The curve intersects the y-axis at 1. So, the y-intercept is 1 (where the x value is 0).

Example 2:

Graph the exponential equation $\mathrm{y}=\frac{1}{3^{\mathrm{x}}}$

Solutions:

Remember to choose a range of values for your x coordinates like -4 to 4 in this case.

So, we need to solve for the value of y. Recall that $\mathrm{y}=\frac{1}{3^{\mathrm{x}}}$

When x = -4

$y=\frac{1}{3^{-4}}$

$\mathrm{y}=3^4$ (note that indices, an exponent with a negative sign gives an inverse of the expression $\frac{1}{3^4}$ )

y = 81

Repeat this step for the values x = -3, -2, -1, 0, 1, 2, 3 and 4 and your answer would be 27, 9, 3, 1, 1/3, 1/9, 1/27 and 1/81.

Now you can complete your table:

A quick look tells you that the y values decrease . This means that there has been decay . Go ahead and plot the graph below:

A graph showing exponential decay for b <1

The following can be understood from the exponential decay graph:

• The curve decreases as the value of x increases.
• Like the growth graph, there is no x-intercept.
• The curve intersects the y-axis at 1 when x is 0. The y-intercept is 1.

Let's plot exponential Graphs with the function for exponential $y=a{b}^x$$y=2\left(3^x\right)$ .

If you follow all the steps which have been explained earlier, your graph should look like this:

Let's look at a combination of the exponential equations in the graph below: $y=3^{x}andy=2\left(3^x\right)$

You will observe that both equations follow the same pattern of increase with no x-intercepts but with y-intercepts = 1 and 2 in the respective equation. This is easily observed when you compare tables of these equations; when x = 0, y = 1 in the first equation and y = 2 in the second equation. Also, we note that the curve $y=3^x$ has been increased by a multiplying factor of 2 in the curve $y=2\left(3^x\right)$ .

We would make another comparison between when a in the equation is positive and negative.$y=a{b}^x$

Let's consider these two equations:

$y=2\left(3^x\right)$

other

$y=-2\left(3^x\right)$

Study the graph below:

From the graph above, we note that for positive values of a the y values are positive (y> 0). However, when a is negative, y gives a negative result (y <0).

Simple applications of exponential growth and decay

In applying the idea of exponential growth and decay, little changes will be made to the general formula:

$y=a{b}^x$

remember that b is the base that represents the growth factor or decay factor. this will be: