- The exponential function is a function in the form of .
- The domain of is the set of all real numbers, and the range is the set of all positive real numbers.
- The line (x-axis) is the horizontal asymptote of .
- If the function is increasing, it is in the form
- If the function is decreasing, it is in the form of .
- The graph of is shown below:
- The graph of is shown below:
Natural Exponential Function, Exponential Growth, and Exponential Decay
- Natural Exponential Function is a function in the form of .
- Euler's number is an irrational number that is approximately equal to 2.71828 or 2.72.
- This function has the same form of graph and properties as those with .
- The inverse of the natural exponential function is the natural logarithmic function.
- A mathematical model of population increase is given by , where is the population after years, is the present population, and is the population growth rate.
- For the doubling time growth model, the equation is , where is the present population, is the population after years, is the doubling, and is the time in years.
- Another form of exponential growth is the concept of compound interest given by the formula where is the initial amount, is the value at the end of years, is the nominal rate, and is the number of conversions.
- The appreciation formula is given by where is the increased value after years, is the initial value, and is the rate of increase.
- The expression is called the growth factor.
- This is modeled by the equation where is the rate of decrease, is Euler's number.
- The half-life decay model is given by where is the half-life.
- The formula for depreciation is given by where is the annual rate of depreciation.
- The expression is called the decay factor.
Transformation of the Graph of Exponential Function
- Transformation of the graph of an exponential function describes its relationship with the parent function .
- Reflection: The graph of is the reflection about the x-axis of the graph of . The graph of is the reflection of the y-axis of the graph of .
- Stretching and Shrinking: In function the effect can be observed in the graph of the parent function . If , the graph shrinks. If , then the graph of is wider.
Vertical and Horizontal Shift:
- For function , the graph of the parent function shifts c units to the right since
- For function , the graph of the parent function shifts c units to the left since .
- For function , the graph of the parent function shifts d units downward since The horizontal asymptote is the line .
- For function , the graph of the parent function shifts d units upward since . The horizontal asymptote is the line .
Exponential Equations and Some Solved Examples
- An exponential equation is an equation involving exponential expressions. The following are some examples. Notice that the exponent has variables.
- Use the One-to-One Property of Exponential Functions in solving exponential equations. Let be a positive real number, with and let and be real numbers. If , then
Example 1. Solve for in the equation .
Thus, we get .
Example 2. Solve the equation
Since there are common bases, we have:
Solved Examples of Exponential Growth and Exponential Decay
Example 1. The population of a particular city in Florida increases according to the exponential model where is in the years. What will the population be after seven years?
Hence, it is predicted to have a total population of 5,000 after seven years.
Example 2. A toxic radioactive substance, Plutonium-230, has a half-life of 24,100 years. Suppose 5 mg of Plutonium was released in a nuclear accident; how much of the 5 mg will remain after 100 years?
Use the exponential decay model,
Hence, about 4.985 mg will be left after 100 years.
- There are two major forms of exponential functions, where (increasing) and where (decreasing).
- The graphs of the exponential functions and show a reflection in the y-axis.
- The graphs of the exponential functions and show a reflection in the x-axis.
- The horizontal asymptote of the exponential function in the form of is the line (x-axis).
- The horizontal asymptote of is the line .
- Using the one-to-one property of exponential functions implies that both expressions must have the same base to solve the equation.
- The base of an exponential function is always positive. Don't be confused with the form .
- An exponential function in the forms or have no x-intercepts. However, the case of has an x-intercept.
- Always use the One-to-One Property of Exponential functions in solving exponential equations.
- In the case of exponential equations wherein we cannot change both sides to similar bases, logarithms are applicable. Some examples of these equations are , , or .
- Keith Madrilejos
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