Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 2 Linear Functions and Equations

2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Equations of Lines ♦ Write the point-slope and slope-intercept forms ♦ Find the intercepts of a line ♦ Write equations for horizontal, parallel, and perpendicular lines ♦ Model data with lines and linear functions (optional) ♦ Use linear regression to model data (optional) 2.1

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Point-Slope Form of the Equation of a Line The line with slope m passing through the point (x 1, y 1 ) has equation y = m(x  x 1 ) + y 1 or y  y 1 = m(x  x 1 ), The point-slope form of the equation of a line.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Find an equation of the line passing through the points (–2, –3) and (1, 3). Plot the points and graph the line by hand. Example: Determining a point- slope form Solution Calculate the slope:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Substitute (1, 3) for (x 1, y 1 ) and 2 for m y = m(x –x 1 ) + y 1 y = 2(x – 1) + 3 Or substitute (–2, –3) for (x 1, y 1 ) and 2 for m y = m(x –x 1 ) + y 1 y = 2(x – (–2)) + (–3) y = 2(x + 2) – 3 Example: Determining a point slope form

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Solution (continued) Here’s the graph: Example: Determining a point- slope form

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Slope-Intercept Form of the Equation of a Line The line with slope m and y-intercept b is given by y = mx + b, y = mx + b, the slope-intercept form of the equation of a line.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Find the slope-intercept form for the line passing through the points (–2, 1) and (2, 3). Example: Finding slope-intercept form Determine m and b in the form y = mx + b Solution Substitute either point to find b, use (2, 3).

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Finding Intercepts To find any x-intercepts, let y = 0 in the equation and solve for x. To find any y-intercepts, let x = 0 in the equation and solve for y. To find any x-intercepts, let y = 0 in the equation and solve for x. To find any y-intercepts, let x = 0 in the equation and solve for y.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Locate the x- and y-intercepts for the line whose equation is 4x + 3y = 6. Use the intercepts to graph the equation. Example: Finding Intercepts To find the x-intercept, let y = 0, solve for x: Solution The x-intercept is 1.5

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 To find the y-intercept, let x = 0, solve for y: The y-intercept is 2. The graph passes through the points (1.5, 0) and (0, 2). Example: Finding Intercepts

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Graph of a constant function f Formula: f (x) = b Horizontal line with slope 0 and y-intercept b. (-3, 3)(3, 3) Horizontal Lines Note that regardless of the value of x, the value of y is always 3.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Vertical Lines Cannot be represented by a function Slope is undefined Equation is: x = k Note that regardless of the value of y, the value of x is always 3. Equation is x = 3 (or x + 0y = 3) Equation of a vertical line is x = k where k is the x-intercept.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Equations of Horizontal and Vertical Lines An equation of the horizontal line with y-intercept b is y = b. An equation of the vertical line with x-intercept k is x = k.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Parallel Lines Two lines with slopes m 1 and m 2, neither of which is vertical, are parallel if and only if their slopes are equal; that is, m 1 = m 2.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Perpendicular Lines Two lines with nonzero slopes m 1 and m 2,, are perpendicular if and only if their slopes have a product of –1; that is, m 1 m 2 = –1.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Perpendicular Lines For perpendicular lines, m 1 and m 2, are negative reciprocals.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Example: Finding perpendicular lines Find the slope-intercept form of the line perpendicular to passing through the point (–2, 1). Graph the lines. The line has slope Solution The negative reciprocal is Use the point-slope form of the line...

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 Example: Finding perpendicular lines

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Example: Modeling investments for cloud computing The table lists the investments in billions of dollars for cloud computing for selected years. (a)Make a scatterplot of the data. (b)Find a formula in point-slope form for a linear function f that models the data. (c)Graph the data and y = f(x) in the same xy-plane. (d) Interpret the slope of the graph of y = f(x). (e) Estimate the investment in cloud computing in Does your answer involve interpolation or extrapolation? Year Investments

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Solution (a)Make a scatterplot of the data. Example: Modeling investments for cloud computing

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 (b)First data point (2005, 26); last data point (2009, 374) Example: Modeling investments for cloud computing

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 (c) Example: Modeling investments for cloud computing

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 24 (d) Interpret the slope of the graph of y = f(x). Slope 87 indicates that investments increased, on average, by $87 billion per year between 2005 and Example: Modeling investments for cloud computing

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 25 (e) Estimate the investment in cloud computing in Does your answer involve interpolation or extrapolation? To estimate the investments in 2014, we can evaluate f(2014). This model predicts an $809 billion investment in cloud computing during This result involves extrapolation because 2014 is “outside” of 2005 and Example: Modeling investments for cloud computing