Many functions containing a variable exponent, are called exponential functions. Formally, any function that can be written in the following form is an exponential function.
Here, the coefficient, is the -intercept, which is sometimes referred to as the initial value. The base, can be interpreted as the constant multiplier. To ensure that is an exponential function, there are restrictions on and
If the coefficient is the function becomes This is a line along and thus, a linear relationship. Therefore, if the function is not exponential.
The initial value, of an exponential function is the number without an exponent. In this case, The constant multiplier, is the number with the exponent. Here,
When the -value increases by the -value is multiplied by Since the -value when is Thus, also lies on the graph of the function. Similarly, the point lies on the graph because These points are shown on the graph.
This process can be repeated until a general form of the graph emerges.
Graph the exponential function and describe its key features
Graph the exponential function using the function rule and describe its key features.
The function has the initial value and the constant multiplier Let us use these values to mark four points on the function's graph.
The function can now be graphed by connecting the points with a smooth curve.
Let us now describe the function's key features.
- First the graph shows a -intercept at
- The function is greater than for all Although the left-end of the graph approaches the -axis it never intersects it. Thus, there is no -intercept.
- As approaches the function continues to approach, but never becomes parallel with, the -axis. Thus, the function increases for all
- Looking at the graph, we can see that the left end approaches and the right end extends upward. Thus, the end behavior of can be written as follows.
Let us show this in the graph.
Interpret the initial value and constant multiplier
In scientists discovered a rare population of Flemish Giant rabbits in a secluded forest. Since then, they've been monitoring the population. During the five years of the study, the number of rabbits could be modeled with the exponential function shown.
Use the graph to write the rule for the function, then interpret its initial value and constant multiplier.
To write an exponential function rule in the form we need the initial value of the function, and the constant multiplier, Notice that the graph starts at This means that is the initial value.
Since we can write the following incomplete function rule. To determine we can use another point on the graph.
Exponential growth occurs when a quantity increases by the same factor over equal intervals of time. This leads to an exponential function, where the independent variable in the exponent, is time. Since the quantity increases over time, the constant multiplier has to be greater than Thus, can be split into two terms, and where is some positive number. The resulting function is called an exponential growth function.
The constant can then be interpreted as the rate of growth, in decimal form. A value of for instance, means that the quantity increases by over every unit of time. As is the case with all exponential functions, is the -coordinate of the -intercept. The base of the power, is known as the growth factor, and since it's greater than the quantity grows faster and faster, without bound.
Exponential decay is very similar to exponential growth. However, instead of the quantity increasing, it decreases by the same factor over equal segments of time. This behavior is also described by an exponential function, However, the constant multiplier is now less than Since it is less than it can be expressed as a subtraction of from where is some positive number less than The resulting function is called an exponential decay function. In this context, the constant can be interpreted as the rate of decay, in decimal form. A value of as an example, means that the quantity decreases by over every unit of time. Same as for an exponential growth function, is the -coordinate of the -intercept. However, since the base is smaller than the quantity decays towards over time.
Find the rule of the exponential growth function
In an ideal environment, bacteria populations grow exponentially and can be modeled with an exponential growth function. The bacteria Lactobacillus acidophilus duplicates about once every hour. A single bacteria cell is placed in an ideal environment. State and interpret the constants and for the growth that will occur. Then, write a function rule describing this growth.
The constant is the initial value of the quantity, in this case the number of bacteria. There was only one bacteria placed in the environment, so is The constant is the rate of growth. Since the bacteria duplicate every hour, the amount of bacteria doubles every hour. This corresponds to an increase by Thus, is Substituting this into the rule of an exponential growth function gives which can be simplified as Since this is an exponential growth function, population will grow faster and faster, without bound. In a real environment, this would not happen, since the available space and nutrition would have to be infinite. At some point, the environment would no longer be ideal, so the growth would slow down or stop.
Graph the exponential growth or decay function
During a time period, the number of carps in a small lake can be modeled by the function where is the time in years. State whether the function shows growth or a decay, and then find the rate of growth or decay, Finally, graph the function.
To begin, let's analyze the given function rule. It's written in the form where is the initial value and is constant multiplier/growth factor. The constant multiplier, is less than so it is a decay factor. Therefore, the function shows decay. Since the decay factor is always equal to we can write the equation. which can be solved for
Thus, the rate of decay is or per year. The initial value is and the constant multiplier is Using this information, we can graph the exponential decay function by plotting some points that lie on and connecting them with a smooth curve.
When money is deposited to a savings account, interest is accrued, often yearly. Different types of interest work in different ways. When the interest earned is then added to the original amount, future interest accrues for a larger amount. This is called compound interest. To calculate the balance on the account at a specific time, an exponential growth function can be used. When the interest is compounded yearly, the balance can be modeled with a function.
In this context, stands for the principal, which is the initial amount of money, and is the interest rate in decimal form. If the interest is not compounded yearly, the function looks a little different.
The constant is the number of times the interest is compounded per year, while is still the annual interest rate. For an account with the principal $ and an annual interest of compounded twice a year, the growth function is:
Notice that this function grows continuously, whereas, in reality, the account balance only increases at the times of compound. Graphing the function together with the actual balance of the account will highlight how it can be used in practice.
Find the monthly interest rate
One savings account, with a principal of $ offers an annual interest rate of compounded twice a year. Find the balance in the account after years. Another savings account with the same principal will have the same balance after years. However, the interest is compounded monthly. Find the interest rate of the second account.
First, we'll find the function rule describing the growth of the first account. It is given that and Substituting these values in the compound interest formula gives Let's simplify this function before continuing.
The account balance is $ after years. Now we can consider the second account. We know the balance of both accounts is equal, at least when both just had their interest compounded. This means that also describes the growth in the second account. However, since the interest in the second account accrues monthly, or times a year, Thus, the exponent in the rule should be By using the equality and the power of a power property, we can rewrite so that it's possible to find the monthly interest rate. Having the exponent means that the base of the power is equal to the monthly growth factor, which we can then use to find the monthly interest rate.
We find an approximate monthly growth factor which corresponds to a rate of growth that is Thus, the monthly interest rate is roughly