1.4 Linear Functions and Slope 1.5 More on Slope Objectives: Calculate slope and write linear equations in different forms. Find slopes of parallel and perpendicular lines. Definition of Slope Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Example: Using the Definition of Slope Find the slope of the line passing through the points (4, –2) and (–1, 5) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 1 Possibilities for Line’s Slope Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Point-Slope Form of the Equation of a Line The point-slope form of the equation of a nonvertical line with slope m that passes through the point (x1, y1) is y y1 m( x x1 ). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Writing an Equation in Point-Slope Form for a Line Write an equation in point-slope form for the line with slope 6 that passes through the point (2, –5). Then solve the equation for y. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 2 Slope-Intercept Form of the Equation of a Line The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is y mx b. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Graphing Using the Slope and y-Intercept 3 Graph the linear function: f ( x) x 1. 5 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b, where b is the y-intercept of the line. The slope of a horizontal line is zero. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 3 Equation of a Vertical Line A vertical line is given by an equation of the form x = a, where a is the x-intercept of the line. The slope of a vertical line is undefined. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Graphing a Horizontal Line Graph y = 3 in the rectangular coordinate system. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 General Form of the Equation of a Line Every line has an equation that can be written in the general form Ax + By + C = 0 where A, B, and C are real numbers, and A and B are not both zero. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 4 Example: Finding the Slope and the y-Intercept Find the slope and the y-intercept of the line whose equation is 3x 6 y 12 0. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Using Intercepts to Graph Ax + By + C = 0 1. Find the x-intercept. Let y = 0 and solve for x. Plot the point containing the x-intercept on the x-axis. 2. Find the y-intercept. Let x = 0 and solve for y. Plot the point containing the y-intercept on the y-axis. 3. Use a straightedge to draw a line through the points containing the intercepts. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Using Intercepts to Graph a Linear Equation Graph using intercepts: 3x 2 y 6 0. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 5 A Summary of the Various Forms of Equations of Lines Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Application Use the data points (317, 57.04) and (354, 57.64) to obtain a linear function that models average global temperature, f(x), for an atmospheric carbon dioxide concentration of x parts per million. Round m to three decimal places and b to one decimal place. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Slope and Parallel Lines 1. If two nonvertical lines are parallel, then they have the same slope. 2. If two distinct nonvertical lines have the same slope, then they are parallel. 3. Two distinct vertical lines, both with undefined slopes, are parallel. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 6 Example: Writing Equations of a Line Parallel to a Given Line Write an equation of the line passing through (–2, 5) and parallel to the line whose equation is y 3 x 1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Slope and Perpendicular Lines 1. If two nonvertical lines are perpendicular, then the product of their slopes is –1. 2. If the product of the slopes of two lines is –1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Writing Equations of a Line Perpendicular to a Given Line Find the slope of any line that is perpendicular to the line whose equation is x 3 y 12 0 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 7 Slope as Rate of Change Slope is defined as the ratio of the change in y to a corresponding change in x. It describes how fast y is changing with respect to x. For a linear function, slope may be interpreted as the rate of change of the dependent variable per unit change in the independent variable. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Slope as Rate of Change In 1990, 9 million adult men in the United States lived alone. In 2008, 14.7 million adult men in the United States lived alone. Use this information to find the slope of the linear function representing adult men living alone in the United States. Express the slope correct to two decimal places and describe what it represents. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 The Average Rate of Change of a Function Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 8 Example: Finding the Average Rate of Change Find the average rate of change for f ( x) x3 x1 = –2 to x2 = 0. Copyright © 2014, 2010, 2007 Pearson Education, Inc. from 25 Average Velocity of an Object Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Finding Average Velocity The distance, s(t), in feet, traveled by a ball rolling down a ramp is given by the function s (t ) 4t 2 where t is the time, in seconds, after the ball is released. Find the ball’s average velocity from t1 = 1 second to t2 = 2 seconds. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 9
© Copyright 2024 Paperzz