Use the data shown in the table at the left, which shows the...

Use the data shown in the table at the left, which shows the number of bacteria present after a certain number of hours. An exponential equation is a nonlinear regression equation of the form $y=a b^{x}$ Use technology to find and graph the exponential equation for the original data. Include the original data in your graph. Note that you can also find this model by solving the equation log $y=m x+b$ from Exercise 38 for $y$

Use the data shown in the table at the left, which shows the number of bacteria present after a certain number of hours.
An exponential equation is a nonlinear regression equation of the form $y=a b^{x}$ Use technology to find and graph the exponential equation for the original data. Include the original data in your graph. Note that you can also find this model by solving the equation log $y=m x+b$ from Exercise 38 for $y$

Key Concepts

Exponential Function

An exponential function is a mathematical relationship expressed in the form y = a · b^x, where 'a' is the initial value and 'b' is the base that represents the growth or decay factor. This type of function models situations where quantities change multiplicatively over equal intervals of time.

Nonlinear Regression

Nonlinear regression is a method used to fit a model to data when the underlying relationship is not linear. In the context of an exponential function, it involves estimating the parameters 'a' and 'b' so that the function y = a · b^x best represents the trend in the data.

Logarithmic Transformation

A logarithmic transformation involves taking the logarithm of both sides of an exponential equation to linearize the model. By transforming y = a · b^x into log(y) = log(a) + x·log(b), one can use linear regression techniques to estimate the parameters, simplifying the process of determining the best fit.

Data Visualization

Data visualization involves creating graphical representations of data to better understand and interpret the underlying patterns and trends. In this context, plotting the original data alongside the graph of the exponential model allows for the assessment of how well the model fits the observed data.

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