Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 3 Equations and Inequalities in Two Variables; Functions
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Equations and Inequalities in Two Variables; Functions 3.1Graphing Linear Equations 3.2The Slope of a Line 3.3The Equation of a Line 3.4Graphing Linear Inequalities 3.5Introduction to Functions and Function Notation CHAPTER 3
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 The Slope of a Line 1.Compare lines with different slopes. 2.Graph equations in slope-intercept form. 3.Find the slope of a line given two points on the line. 3.2
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Example Graph each of the following on the same grid. y = xy = 3xy = 4x Solution Complete a table of values. If x isy = xy = 3xy = 4x y = x y = 3x y = x y = 3x y = x y = 4x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Graph each of the following on the same grid. Solution Complete a table of values. If x is y = x 0000 11 1 22 2
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 If the coefficient of m increases, the graphs get steeper. Because the coefficient m affects how steep a line is, m is called the slope of the line. If a slope of the line is a fraction between 0 and 1, then the smaller the fraction is, the less inclined or flatter the line gets.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Slope: The ratio of the vertical change (change in y) to the horizontal change (change in x) between any two points on a line between those points.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example For the equation determine the slope and the y-intercept. Then graph the equation. Solution m = y-intercept: (0, 3) Plot the y-intercept and then use the slope to find other points. rise 2 run 3 (3, 1)
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Graphs of Equations in Slope-Intercept Form The graph of an equation in the form y = mx + b, (slope-intercept form) is a line with slope m and y- intercept (0, b). The following rules indicate how m affects the graph. If m > 0, the line slants upward from left to right. If m < 0, the line slants downward from left to right. The greater the absolute value of m, the steeper the line. m > 0 m < 0
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example For the equation 2x + 5y = 20, determine the slope and the y-intercept. Then graph the equation. Solution Write the equation in slope-intercept form by isolating y. 2x + 5y = 20 5y = 2x 20 The slope is and the y-intercept is (0, –4).
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued We begin at (0, –4) and then rise 2 and run 5. m = y-intercept: (0, –4) rise 2 run 5
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 The Slope Formula Given two points (x 1, y 1 ) and (x 2, y 2 ), where x 2 x 1, the slope of the line connecting the two points is given by the formula Rise: y 2 – y 1 Run: x 2 – x 1 (x 1, y 1 ) (x2, y 2 )
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Find the slope of the line connecting (4, 6) and (−2, 8). Solution Example Using, replace the variables with their corresponding values and then simplify.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Graph the line connecting the given points and find its slope. (3, 8) and (−2, 8) Solution Example Because the y-coordinates are the same, the graphs is a horizontal line.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Solution Example Because the x-coordinates are the same, the graphs is a vertical line. Graph the line connecting the given points and find its slope. (6, 1) and (6, −4)
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Slopes of Horizontal and Vertical Lines Two points with different x-coordinates and the same y- coordinates, (x 1, c) and (x 2, c), will form a horizontal line with slope 0 and equation y = c. Two points with the same x-coordinates and different y- coordinates, (c, y 1 ) and (c, y 2 ), will form a vertical line with undefined slope and equation x = c.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Example The following graph shows the hourly wage earned by Henry each of the five years after his hire date. An analysis determines that the red line reasonably describes the trend shown in the data. Find the slope of that line. Understand We are to find the slope on a line that passes through or near a set of data points.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 continued Execute Plan To find the slope of the line, we use the slope formula with two data points that are on the line. We will use (3, 15) and (5, 18). Answer The slope of the line is, which means that Henry’s wage increased about $1.50 per hour each year. Check We could use a different pair of data points on the line and see if we get the same slope. We will leave this to the viewer.