a. What function, linear or exponential, do you think would be the best choice to model this data? Explain why you think your choice is the best-fit. b. Explain how to determine which function is the best choice mathematically, without using a graph. Demonstrate your method. c. Think of an example of at least 4 data points that would be best modeled by the function you did NOT choose in part a. Explain how you set up the data so that it worked with this type of function.

You should be familiar with how to graph three very important types of equations:

Linear equations in slope-intercept form:

y=mx+b

Exponential equations of the form:

y=a(b)x

Quadratic equations in standard form:

y=ax2+bx+c

In real-world applications, the function that describes a physical situation is not always given. Finding the function is an important part of solving problems. For example, scientific data such as observations of planetary motion are often collected as a set of measurements presented in a table. One job for a scientist is to figure out which function best fits the data.

Using Differences to Determine the Model

By finding the differences between dependent values, you can determine the degree of the model for data given as ordered pairs.

If the first difference is the same value, the model will be linear.

If the second difference is the same value, the model will be quadratic.

If the number of times the difference has been taken before finding repeated values exceeds five, the model may be exponential or some other special equation.

Determine which model to use given the following tables of values:

The first difference (the difference between any two successive output values) is the same value (3). This means that this data can be modeled using a linear regression line.

The equation to represent this data is

y=3x+2.

This is a quadratic model because the second differences are the differences that have the same value (4). Note that when you compare the difference of the

y-

values, you must make sure that you examine entries for which the

x-

values increase by the same amount each time.

Using Ratios to Determine the Model

Finding the differences involves subtracting the dependent values leading to the degree of the model. By taking the ratio of the values, one can determine whether the model is exponential.

If the ratio of dependent values is the same, then the data is modeled by an exponential equation.

Find the model that represents the following table of values:

Note that the ratio of values is the same between each set of numbers. This is an exponential equation.

The equation to represent this data is