1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Lines SECTION Find the slope of a line. Write the point-slope form of the equation of a line. Write the slope-intercept form of the equation of a line. Recognize the equations of horizontal and vertical lines. Recognize the general form of the equation of a line. Find equations of parallel and perpendicular lines. Use linear regression

3 © 2010 Pearson Education, Inc. All rights reserved Because the graphs of first degree equations in two variables are straight lines, these equations are called linear equations. We measure the “steepness” of a line by a number called its slope. Definitions

4 © 2010 Pearson Education, Inc. All rights reserved Definitions The rise is the change in y-coordinates between the points and the run is the corresponding change in the x-coordinates.

5 © 2010 Pearson Education, Inc. All rights reserved SLOPE OF A LINE The slope of a nonvertical line that passes through the points P(x 1, y 1 ) and Q(x 2, y 2 ) is denoted by m and is defined by The slope of a vertical line is undefined.

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding and Interpreting the Slope of a Line Sketch the graph of the line that passes through the points P(1, –1) and Q(3, 3). Find and interpret the slope of the line. Solution The graph of the line passing through the points P(1, –1) and Q(3, 3) is sketched here.

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding and Interpreting the Slope of a Line The slope m of the line through P(1, –1) and Q(3, 3) is given by Solution continued

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding and Interpreting the Slope of a Line Solution continued A slope of 2 means that the value of y increases two units for every one unit increase in the value of x. Interpretation

9 © 2010 Pearson Education, Inc. All rights reserved MAIN FACTS ABOUT SLOPES OF LINES 1.Scanning graphs from left to right, lines with positive slopes rise and lines with negative slopes fall. 2.The greater the absolute value of the slope, the steeper the line. 3.The slope of a vertical line is undefined. 4.The slope of a horizontal line is 0.

10 © 2010 Pearson Education, Inc. All rights reserved POINT–SLOPE FORM OF THE EQUATION OF A LINE If a line has slope m and passes through the point (x 1, y 1 ), then the point-slope form of an equation of the line is

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding an Equation of a Line with Given Point and Slope Find the point-slope form of the equation of the line passing through the point (1, –2) and with slope m = 3. Then solve for y. Solution

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding an Equation of a Line Passing Through Two Given Points Find the point-slope form of the equation of the line l passing through the points (–2, 1) and (3, 7). Then solve for y. Solution First, find the slope.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding an Equation of a Line Passing Through Two Given Points Solution continued

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Find the point-slope form of the equation of the line with slope m and y-intercept b. Then solve for y. Finding an Equation of a Line with a Given Slope and y-intercept Solution The line passes through (0, b).

15 © 2010 Pearson Education, Inc. All rights reserved SLOPE–INTERCEPT FORM OF THE EQUATION OF A LINE The slope-intercept form of the equation of the line with slope m and y-intercept b is

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Use the slope to find a point on the line whose equation is different from the y- intercept, then graph the line. Graphing by Using the Slope and y -intercept Solution The equation is in the slope- intercept form with slope and y-intercept 2. To sketch the graph, find two points on the line and draw a line through the two points.

17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing by Using the Slope and y -intercept Because let 2 be the rise and 3 be the run. From the point (0, 2) move 3 units to the right and 2 units up. Solution continued Use the y-intercept as one of the points; then use the slope to locate a second point. The line we want joins the points.

18 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL AND VERTICAL LINES An equation of a horizontal line through (h, k) is y = k. An equation of a vertical line through (h, k) is x = h.

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Discuss the graph of each equation in the xy-plane. Recognizing Horizontal and Vertical Lines Solution The equation y = 2 may be considered as an equation in two variables x and y by writing 0 ∙ x + y = 2. Any ordered pair of the form (x, 2) is a solution of the equation. The graph of y = 2 is a line parallel to the x-axis and 2 units above it with a slope of 0.

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Recognizing Horizontal and Vertical Lines Solution continued

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Recognizing Horizontal and Vertical Lines Solution continued The equation may be written as x + 0 ∙ y = 4. Any ordered pair of the form (4, y) is a solution of the equation. The graph of x = 4 is a line parallel to the y-axis and 4 units to the right of it. The slope of a vertical line is undefined.

22 © 2010 Pearson Education, Inc. All rights reserved GENERAL FORM OF THE EQUATION OF A LINE The graph of every linear equation ax + by + c = 0, where a, b, and c are constants and not both a and b are zero, is a line. The equation ax + by + c = 0 is called the general form of the equation of a line.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Linear Equation Find the slope, y-intercept, and x-intercept of the line with equation 3x – 4y +12 = 0. Then sketch the graph. Solution The y-intercept is 3. Set y = 0 and solve for x: 3x + 12 = 0. The x-intercept is – 4. The slope is

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Linear Equation Solution continued

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Inferring Height from the Femur The height (H) of a human male is related to the length (x) of his femur by the formula where all measurements are in centimeters. The femur of Wild Bill Longley was measured to be between 45 and 46 centimeters. Estimate the height of Wild Bill.

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Inferring Height from the Femur Substitute x = 45 and x = 46 Wild Bill’s height was between 182 and cm. (1 in = 2.54 cm) Solution Wild Bill was approximately 6 feet tall.

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Linear Depreciation The graph in the figure represents the value of a small plane over a ten-year period. Each point (t, y) on the graph gives the plane’s value after t years.

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Linear Depreciation a. What does the y-intercept represent? b. What is the value of the plane after ten years? c. How much does the plane depreciate in value each year? d. Write a linear equation for the value of the plane after t years for 0  t  10. e. What does the slope found in part (d) represent?

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Linear Depreciation a. The y-intercept represents the purchase value of the plane, $140,000. b. The graph point (10, 90) gives the value of the plane after ten years as $90,000. c. The value of the plane decreased from $140,000 to $90,000 over ten years. This is per year. Solution

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Linear Depreciation d. Using the points (0, 140) and (10, 90), we find that the slope of the depreciation line is Then Solution continued e. The slope m = −5 gives the plane’s yearly depreciation of $5000 per year.

31 © 2010 Pearson Education, Inc. All rights reserved PARALLEL AND PERPENDICULAR LINES Let l 1 and l 2 be two distinct lines with slopes m 1 and m 2, respectively. Then l 1 is parallel to l 2 if and only if m 1 = m 2. l 1 is perpendicular l 2 to if and only if m 1 ∙ m 2 = –1. Any two vertical lines are parallel, and any horizontal line is perpendicular to any vertical line.

32 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Let l: ax + by + c = 0. Find the equation of each line through the point (x 1, y 1 ): (a) l 1 parallel to l (b) l 2 perpendicular to l Step 1 Find slope m of l. EXAMPLE 10 Finding Equations of Parallel and Perpendicular Lines EXAMPLE Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8): (a) l 1 parallel to l (b) l 2 perpendicular to l 1. l : The slope of l is

33 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Let l: ax + by + c = 0. Find the equation of each line through the point (x 1, y 1 ): (a) l 1 parallel to l (b) l 2 perpendicular to l Step 2 Write slope of l 1 and l 2. The slope m 1 of l 1 is m. The slope m 2 of l 2 is EXAMPLE 10 Finding Equations of Parallel and Perpendicular Lines EXAMPLE Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8): (a) l 1 parallel to l (b) l 2 perpendicular to l

34 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Let l: ax + by + c = 0. Find the equation of each line through the point (x 1, y 1 ): (a) l 1 parallel to l (b) l 2 perpendicular to l Step 3 Write the equations of l 1 and l 2. Use point-slope form to write equations of l 1 and l 2. Simplify to write equations in the requested form. EXAMPLE 10 Finding Equations of Parallel and Perpendicular Lines EXAMPLE Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8): (a) l 1 parallel to l (b) l 2 perpendicular to l l1:l1: l2:l2:

35 © 2010 Pearson Education, Inc. All rights reserved Modeling Data Using Linear Regression In many applications, the way in which one quantity depends on another can be modeled by a linear function. One technique for finding this linear function is called linear regression. The line used to model the data is called the regression line or the least-squares line.

36 © 2010 Pearson Education, Inc. All rights reserved Modeling Data Using Linear Regression

37 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 11 Predictions Using Linear Regression Use the equation y = 0.38x to predict the number of registered motorcycles in the United States for Because 2007 is seven years after 2000, we set x = 7; then y = 0.38(7) = 7. So according to the regression prediction, 7 million motorcycles were registered in the United States for Solution