Equations of Lines and Modeling Section 1.4 Equations of Lines and Modeling

Objectives Determine equations of lines. Given the equations of two lines, determine whether their graphs are parallel or perpendicular. Model a set of data with a linear function. Fit a regression line to a set of data. Then use the linear model to make predictions.

Slope-Intercept Equation Recall the slope-intercept equation y = mx + b or f (x) = mx + b. If we know the slope and the y-intercept of a line, we can find an equation of the line using the slope-intercept equation.

Example A line has slope and y-intercept (0, 16). Find an equation of the line. We use the slope-intercept equation and substitute for m and 16 for b:

Example A line has slope and contains the point (–3, 6). Find an equation of the line. We use the slope-intercept equation and substitute for m:

Example (continued) Using the point (3, 6), we substitute –3 for x and 6 for y, then solve for b. Equation of the line is

Point-Slope Equation The point-slope equation of the line with slope m passing through (x1, y1) is y  y1 = m(x  x1).

Example Find the equation of the line containing the points (2, 3) and (1, 4). First determine the slope: Using the point-slope equation, substitute 7 for m and either of the points for (x1, y1):

Parallel Lines Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

Perpendicular Lines Two lines with slopes m1 and m2 are perpendicular if and only if the product of their slopes is 1: m1m2 = 1.

Perpendicular Lines Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).

Example Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. a) y + 2 = 5x, 5y + x =  15 Solve each equation for y. The slopes are negative reciprocals, and their product is −1, so the lines are perpendicular.

Example (continued) Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. b) 2y + 4x = 8, 5 + 2x = –y Solve each equation for y: Since the slopes are the same and y-intercepts are different, the lines are parallel.

Example (continued) Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. c) 2x +1 = y, y + 3x = 4 Solve each equation for y. Since the slopes are not the same and their product is not −1, the lines are neither parallel nor perpendicular.

Example Write equations of the lines (a) parallel to and (b) perpendicular to the graph of the line 4y – x = 20 and containing the point (2,  3). Solve the equation for y: So the slope of this line is

Example (continued) The line parallel to the given line will have the same slope, 1/4 . We use either the slope-intercept or point-slope equation for the line. Substitute 1/4 for m and use the point (2,  3) and solve the equation for y.

Example (continued) (b) The slope of the perpendicular line is the opposite of the reciprocal of 1/4, or – 4. Use the point-slope equation, substitute – 4 for m and use the point (2, –3) and solve the equation.

Mathematical Models When a real-world problem can be described in mathematical language, we have a mathematical model. The mathematical model gives results that allow one to predict what will happen in that real-world situation. If the predictions are inaccurate or the results of experimentation do not conform to the model, the model must be changed or discarded. Mathematical modeling can be an ongoing process.

Curve Fitting In general, we try to find a function that fits, as well as possible, observations (data), theoretical reasoning, and common sense. We call this curve fitting; it is one aspect of mathematical modeling. Let’s examine some data and related graphs, or scatter plots and determine whether a linear function seems to fit the data.

Example The cost of a 30-sec Super Bowl commercial has increased $1.6 million from 2010 to 2015. Model the data in the table below with a linear function. Then, estimate the cost of a 30-sec commercial in 2018.

Example (continued) Choose any two data points to determine the equation. Let’s use (1, 3.1) and (4, 4.0). First, find the slope. Substitute 0.3 for m and either of the points. We’ll use (1, 3.1) in the point-slope equation.

Example (continued) Where x is number of years after 2010 and y is in millions of dollars. Now we can estimate the cost of a 30-sec commercial in 2018 by substituting 8 (2018 – 2010 = 8) for x in the model: We estimate that the cost of commercial will be $5.2 million in 2018.

Linear Regression Linear regression is a procedure that can be used to model a set of data using a linear function. Consider the data on the cost of a 30-sec Super Bowl commercial given previously. Then use the function to predict the cost of a 30-sec commercial in 2018. We can fit a regression line of the form y = mx + b to the data using the LINEAR REGRESSION feature on a graphing calculator.

Example Fit a regression line to the data given in the table. Then use the function to predict the cost of a 30-sec commercial in 2018. Enter the data in lists on the calculator. The independent variables or x values are entered into L1 and the dependent variables or y values into L2. The calculator can then create a scatterplot.

Example (continued) Here are screen shots of the calculator showing Lists 1 and 2 and the scatterplot of the data.

Example (continued) Here are screen shots of selecting the LINEAR REGRESSION feature from the STAT CALC menu. From the screen in the middle, we find the linear equation that best models the data is

Example (continued) Substitute 8 into the regression equation: Using this model, the cost is predicted to be about $5.36 million in 2018.

The Correlation Coefficient On some graphing calculators with the DIAGNOSTIC feature turned on, a constant r between −1 and 1, called the coefficient of linear correlation, appears with the equation of the regression line. Used to describe the strength of the linear relationship between x and y. The closer | r | is to 1, the better the correlation. A positive value of r indicates that the regression line has a positive slope, and a negative value of r indicates that the regression line has a negative slope.

The Correlation Coefficient The following scatterplots summarize the interpretation of a correlation coefficient.

The Correlation Coefficient The following scatterplots summarize the interpretation of a correlation coefficient.