# How to Write an Exponential Function: Word Problems

Any function in the form $$y=ab^x$$ is called an exponential function. Unlike linear functions where the growth rate is constant, exponential functions are characterized by the fact that the growth rate of the function is proportional to the function's current value. In this article, the method of writing an exponential function is explained step by step.

## A step-by-step guide to Writing exponential function: word problems

Any function in the form $$y=ab^x$$ is called an exponential function where $$a$$ and $$b$$ are fixed and $$x$$ is an independent variable:

Variable $$y$$: represents the output value

$$a$$ represents the initial value

$$b$$ represents the common ratio

$$x$$ represents the number of times the original value has been multiplied by the common ratio

Unlike linear functions where the growth rate is constant, exponential functions are characterized by the fact that the growth rate of the function is proportional to the function’s current value and are used to model growth and decay processes, such as population growth, radioactive decay, and compound interest.

Follow the step-by-step procedure below to write an exponential function.

Step $$1$$: Analyzing the problem and identifying the variables. At this stage, you should specify what the question is about and what the variables of the problem are.

Step $$2$$: Using the variables, write the exponential function in the form of $$y=ab^x$$.

Step $$3$$: Determine the initial value ($$a$$) according to the information of the problem. $$a$$ shows the value of the variable at the beginning of the time period.

Step $$4$$: In this step, you must specify the common ratio ($$b$$). $$b$$ is the coefficient by which the variable is multiplied in each time period.

Step $$5$$: Specify the $$x$$ variable that represents the number of time periods that have passed. For example, if in the problem it is said that a certain amount will be multiplied by $$4$$ every month and after $$5$$ months you will be asked for the amount, $$x=5$$.

Step $$6$$: Place the values of $$a$$, $$b$$ and $$x$$ in the exponential function and simplify it.

Step $$7$$: Check your work by inserting $$x$$ values and seeing if they match the problem statement

Step $$8$$: Write the final answer in one sentence

### Writing exponential function: word problems-Example 1:

If a bank’s profits start at $$60,000$$ and increase by $$25%$$ each year, how much will the profits be after $$4$$ years?

Solution:

This problem is solved by using the exponential function. In the said problem, the profit of the company starts from $$60,000$$ dollars and increases by $$25%$$ every year, the profit is multiplied by $$1.2$$ every year. With this information, we write the exponential function:

$$y = 60000 * 1.25^x$$

In this function, $$x$$ is the number of past years and $$y$$ represents the profit after $$x$$ years.

The profit amount after $$4$$ years is obtained by inserting $$x = 4$$ in the function: $$y = 40000 * 1.25^4 = 60000 * 1.25 * 1.25 * 1.25 * 1.25 = 146484$$

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