Exponential functions are solutions to the simplest types of dynamical systems. For example, an exponential function arises in simple models of bacteria growth.
An exponential function can describe growth or decay. The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay. The function machine metaphor is useful for introducing parameters into a function. We could capture both functions using a single function machine but dials to represent parameters influencing how the machine works.
This returns an equation of the form. Given a set of data, perform logistic regression using a graphing utility. Mobile telephone service has increased rapidly in America since the mid s. Today, almost all residents have cellular service. Figure shows the percentage of Americans with cellular service between the years and [3].
Next, graph the model in the same window as shown in Figure the scatterplot to verify it is a good fit:. To approximate the percentage of Americans with cellular service in the year , substitute for the in the model and solve for.
Figure shows the population, in thousands, of harbor seals in the Wadden Sea over the years to Access this online resource for additional instruction and practice with exponential function models.
Visit this website for additional practice questions from Learningpod. What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit. Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.
What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense? What is regression analysis? Describe the process of performing regression analysis on a graphing utility. Regression analysis is the process of finding an equation that best fits a given set of data points. The shape of the data points on the scatter graph can help determine which regression feature to use. What might a scatterplot of data points look like if it were best described by a logarithmic model?
What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation? The y -intercept on the graph of a logistic equation corresponds to the initial population for the population model. For the following exercises, match the given function of best fit with the appropriate scatterplot in Figure through Figure.
Answer using the letter beneath the matching graph. To the nearest whole number, what is the initial value of a population modeled by the logistic equation What is the carrying capacity?
Rewrite the exponential model as an equivalent model with base Express the exponent to four significant digits.
A logarithmic model is given by the equation To the nearest hundredth, for what value of does. A logistic model is given by the equation To the nearest hundredth, for what value of t does.
What is the y -intercept on the graph of the logistic model given in the previous exercise? For the following exercises, use this scenario: The population of a koi pond over months is modeled by the function. Graph the population model to show the population over a span of years. How many months will it take before there are koi in the pond? Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.
For the following exercises, use this scenario: The population of an endangered species habitat for wolves is modeled by the function where is given in years. How many wolves will the habitat have after years? How many years will it take before there are wolves in the habitat? Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity. For the following exercises, refer to Figure. Use the regression feature to find an exponential function that best fits the data in the table.
Write the exponential function as an exponential equation with base. Use the intersect feature to find the value of for which. Use the LOGarithm option of the REGression feature to find a logarithmic function of the form that best fits the data in the table. Use the logarithmic function to find the value of the function when. To the nearest whole number, what is the predicted carrying capacity of the model?
Use the intersect feature to find the value of for which the model reaches half its carrying capacity. Recall that the general form of a logistic equation for a population is given by such that the initial population at time is Show algebraically that. Working with the left side of the equation, we see that it can be rewritten as. Working with the right side of the equation we show that it can also be rewritten as But first note that when Therefore,.
Use a graphing utility to find an exponential regression formula and a logarithmic regression formula for the points and Round all numbers to 6 decimal places. Graph the points and both formulas along with the line on the same axis. Make a conjecture about the relationship of the regression formulas. Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary. First rewrite the exponential with base e : Then test to verify that taking rounding error into consideration:.
Find the inverse function for the logistic function Show all steps. Use the result from the previous exercise to graph the logistic model along with its inverse on the same axis. What are the intercepts and asymptotes of each function? Determine whether the function represents exponential growth, exponential decay, or neither. The population of a herd of deer is represented by the function where is given in years.
To the nearest whole number, what will the herd population be after years? Find an exponential equation that passes through the points and. Determine whether Figure could represent a function that is linear, exponential, or neither.
If it appears to be exponential, find a function that passes through the points. What will the account be worth in years? To the nearest dollar, how much will she need to invest in an account now with APR, compounded daily, in order to reach her goal in years? Does the equation represent continuous growth, continuous decay, or neither?
Suppose an investment account is opened with an initial deposit of earning interest, compounded continuously. How much will the account be worth after years? Graph the function State the domain and range and give the y -intercept. Graph the function and its reflection about the y -axis on the same axes, and give the y -intercept. The graph of is reflected about the y -axis and stretched vertically by a factor of What is the equation of the new function, State its y -intercept, domain, and range.
The graph below shows transformations of the graph of What is the equation for the transformation? Rewrite as an equivalent exponential equation. Rewrite as an equivalent logarithmic equation. Solve for x if by converting to exponential form. Evaluate without using a calculator. Evaluate using a calculator. Round to the nearest thousandth. Graph the function. State the domain, vertical asymptote, and end behavior of the function.
Domain: Vertical asymptote: End behavior: as and as. Rewrite in expanded form. Rewrite in compact form. Rewrite as a product. Rewrite as a single logarithm. Use properties of logarithms to expand. Condense the expression to a single logarithm. Rewrite to base. Rewrite as a logarithm. Then apply the change of base formula to solve for using the common log.
Solve by rewriting each side with a common base. Use logarithms to find the exact solution for If there is no solution, write no solution. Find the exact solution for. If there is no solution, write no solution. Skip to main content. Fitting Exponential Models to Data. Search for:. As x increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.
As x increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x -axis. In other words, the outputs never become equal to or less than zero. A General Note: Exponential Regression Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
How To: Given a set of data, perform exponential regression using a graphing utility. Clear any existing data from the lists. List the input values in the L1 column. List the output values in the L2 column.
Use ZOOM [9] to adjust axes to fit the data.
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