Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 3 Equations and Inequalities in Two Variables; Functions.

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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 3.2The Slope of a Line 3.3The Equation of a Line 3.5Introduction to Functions and Function Notation CHAPTER 3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 The Equation of a Line 1.Use slope-intercept form to write the equation of a line. 2.Use point-slope form to write the equation of a line. 3.Write the equation of a line parallel to a given line. 4.Write the equation of a line perpendicular to a given line. 3.3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 If we are given the y-intercept (0, b) of a line, to write the equation of the line, we will need either the slope or another point so that we can calculate the slope.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example A line with a slope of 4 crosses the y-axis at the point (0, 5). Write the equation of the line in slope-intercept form. Solution Use y = mx + b m = 4 b = 5 from the point given (0, 5) which is the y-intercept. y = 4x + 5

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Equation of a Line Given Its y-intercept To write the equation of a line given its y-intercept, (0, b), and its slope, m, use the slope-intercept form of the equation, y = mx + b. If given a second point and not the slope, calculate the slope using then use y = mx + b.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Example Write the equation of the line passing through (0,  6) and (3, 6) in slope-intercept form. Solution The y-intercept is (0,  6). Find the slope: y = mx + b y = 4x  6

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 You can use the point-slope form to write the equation of a line given any two points on the line. Using the Point-Slope Form of the Equation of a Line To write the equation of a line given its slope and any point, (x 1, y 1 ), on the line, use the point-slope form of the equation of a line, y – y 1 = m(x – x 1 ). If given a second point (x 2, y 2 ), and not the slope, we first calculate the slope using then use y – y 1 = m(x – x 1 ).

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Write the equation of a line with a slope of 5 that passes through the point (3, 12). Write the equation in slope-intercept form. Solution Begin with the point-slope formula. Replace m = 5, x 1 = 3, y 1 = 12 y – y 1 = m(x – x 1 ) y – 12 = 5(x – 3) y – 12 = 5x – 15 Simplify. y = 5x – 3 Add 15 to both sides to isolate y.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Write the equation of a line passing through the points (4,  2) and (  4, 4). Write the equation in slope- intercept form. Solution Calculate the slope. Use the point-slope form, then isolate y to write the slope-intercept form.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Points (4,  2) and (  4, 4), y – y 1 = m(x – x 1 ) Replace m with -3/4, x 1 with 4 and y 1 with -2. Simplify. Subtract 2 from both sides to isolate y.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Equations can also be written in standard form, which is Ax + By = C, where A, B, and C are real numbers. It should also be written so that the x-term is first and with a positive coefficient and, if possible, A, B, and C are all integers.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example A line connects the points (2, 6) and (–4, 3). Write the equation of the line in the form Ax + By = C, where A, B, and C are integers and A > 0. Solution Find the slope: Use point-slope form: y – y 1 = m(x – x 1 ) Distribute to clear ( ).

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 continued Multiply both sides by the LCD, 2. Subtract x from both sides to get x and y together. Add 12 to both sides to get the constant terms together. Multiply by –1 so that the coefficient of x is positive.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Parallel Lines Nonvertical parallel lines have equal slopes and different y-intercepts. Vertical lines are parallel. y = 2x + 1 y = 2x – 3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Example Write the equation of the line in slope-intercept form that passes through (1, –5) and parallel to the graph of y = –3x + 4. Solution In y = –3x + 4, the slope is –3, so the slope of the line parallel will also be –3. Use point-slope form. y – y 1 = m(x – x 1 ) y – (  5) = –3(x – 1) y + 5 = –3x + 3 y = –3x – 2 y 1 =  5, x 1 = 1 and m = –3 Simplify. Subtract 5 from both sides to isolate y.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Perpendicular Lines The slope of a line perpendicular to a line with a slope of will be Horizontal and vertical lines are perpendicular.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 Example Write the equation of a line in standard form that passes through (7, 1) and is perpendicular to the graph of 7x – 2y = –2. Solution Determine the slope of the line 7x – 2y = –2. Slope of perpendicular line:

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19 continued slope = ; point (7, 1) y – y 1 = m(x – x 1 ) Multiply both sides by 7, simplify and distribute. Rearrange the x and y terms, and add 7 to both sides to put the equation in Standard Form.