Jay Abramson; Sharon North

Key Concepts

4.1 Linear Functions

Linear functions can be represented in words, function notation, tabular form, and graphical form. See Example 1.
An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. See Example 2.
Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See Example 3 and Example 4.
An equation for a linear function can be written from a graph. See Example 5.
The equation for a linear function can be written if the slope $m$ and initial value $b$ are known. See Example 6 and Example 7.
A linear function can be used to solve real-world problems given information in different forms. See Example 8, Example 9, and Example 10.
Linear functions can be graphed by plotting points or by using the y-intercept and slope. See Example 11 and Example 12.
Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See Example 13.
The equation for a linear function can be written by interpreting the graph. See Example 14.
The x-intercept is the point at which the graph of a linear function crosses the x-axis. See Example 15.
Horizontal lines are written in the form, $f (x) = b .$ See Example 16.
Vertical lines are written in the form, $x = b .$ See Example 17.
Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See Example 18.
A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation, $f (x) = m x + b,$ and using the $b$ that results. Similarly, the point-slope form of an equation can also be used. See Example 19.
A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See Example 20 and Example 21.

4.2 Modeling with Linear Functions

We can use the same problem strategies that we would use for any type of function.
When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See Example 1.
Draw a diagram, where appropriate. See Example 2 and Example 3.
Check for reasonableness of the answer.
Linear models may be built by identifying or calculating the slope and using the y-intercept.
- The x-intercept may be found by setting $y = 0,$ which is setting the expression $m x + b$ equal to 0.
- The point of intersection of a system of linear equations is the point where the x- and y-values are the same. See Example 4.
- A graph of the system may be used to identify the points where one line falls below (or above) the other line.

4.3 Fitting Linear Models to Data

Scatter plots show the relationship between two sets of data. See Example 1.
Scatter plots may represent linear or non-linear models.
The line of best fit may be estimated or calculated, using a calculator or statistical software. See Example 2.
Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See Example 3.
The correlation coefficient, $r,$ indicates the degree of linear relationship between data. See Example 4.
A regression line best fits the data. See Example 5.
The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See Example 6.