# Linear Interpolation and Extrapolation

## Linear interpolation helps use the building blocks of geometry to solve algebraic unknowns. Work to identify the values between and beyond known points.

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Making Predictions using Interpolation and Extrapolation

### Learning Goals

I will be able to identify a trend in a scatterplot and use that to predict other values for a given relationship.

Suppose that you've plotted a number of data points on a coordinate plane, with the \begin{align*}x-\end{align*}coordinate of each data point representing the number of months since you've planted a tree, and the \begin{align*}y-\end{align*} coordinate of each point representing the tree's height in meters. If you have data for 1 month, 2 months, 3 months, and 4 months, do you think you could guess what the tree's height will be at 5 months? How about what the tree's height was at 2.5 months? After completing this Concept, you'll be able make these types of guesses by using linear interpolation and linear extrapolation.

### Predicting with Linear Models

Numerical information appears in all areas of life. You can find it in newspapers, in magazines, in journals, on the television, or on the Internet. In the last Concept, you saw how to find the equation of a line of best fit. Using a line of best fit is a good method if the relationship between the dependent and independent variables is linear. Not all data fits a straight line, though. This Concept will show other methods to help estimate data values. These methods are useful in both linear and non-linear relationships.

Linear Interpolation

Linear interpolation is useful when looking for a value between given data points. It can be considered as “filling in the gaps” of a table of data.

We can find our estimate on the line of best fit within the points already plotted.

Linear Extrapolation

Linear extrapolation can help us estimate values that are either higher or lower than the values in the data set. Think of this as “the long-term estimate” of the data.

We will have to extend the line of best fit to extrapolate.

Video Review

Video Example 1

Video Example 2

### Guided Practice

The Center for Disease Control (CDC) has the following information regarding the percentage of pregnant women smokers organized by year. Estimate the percentage of pregnant women that were smoking in the year 1998.

Percent of Pregnant Women Smokers by Year
Year Percent
1990 18.4
1991 17.7
1992 16.9
1993 15.8
1994 14.6
1995 13.9
1996 13.6
2000 12.2
2002 11.4
2003 10.4
2004 10.2

Percent of Pregnant Women Smokers by Year

Solution:

We want to use the information close to 1998 to interpolate the data. We do this by drawing a line of best fit. We can then use that line of best fit to approximate the percentage of women smokers in 1998.

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