2.1 Lines and Slopes The slope of the line through the distinct points ( x1 , y1 ) and ( x2 , y2 ) Change in y Rise y2 − y1 = = is Change in x Run x2 − x1 where x2 − x1 ≠ 0. Example 1: Find the Slope Find the slope of the line passing through the pair of points (2,1) and (3, 4). Solution Let ( x1 , y1 ) = (2,1) and ( x2 , y2 ) = (3, 4). y2 − y1 4 − 1 3 Slope = m = = = 3. = x2 − x1 3 − 2 1 Possibilities for a Line’s Slope Positive Slope Negative Slope m>0 Line rises from left to right. m<0 Line falls from left to right. Possibilities for a Line’s Slope Zero Slope m=0 Line is horizontal. Undefined Slope m is undefined. Line is vertical. Example 2: Find the Slope Find the slope of the line passing through the pair of points (−1,3) and (2, 4) or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. Solution Let ( x1 , y1 ) = (−1,3) and ( x2 , y2 ) = (2, 4). 4−3 1 y2 − y1 = = Slope = m = x2 − x1 2 − (−1) 3 The slope is positive, and the line rises from left to right. Practice Exercise Find the slope of the line passing through the points (4, −1) and (3, −1) or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. Answer The slope m is zero. Thus, the line is a horizontal line. Point-slope Form of the Equation of a Line The point-slope equation of a nonvertical line of slope m that passes through the point ( x1 , y1 ) is y − y1 = m( x − x1 ). Example 3: Writing the PointSlope Equation of a Line Write the point-slope form of the equation of the line passing through (1,3) with a slope of 4. Then slove the equation for y. Solution We use the point-slope equation of a line with m = 4, x1 = 1, and y1 = 3. m = 4, x1 = 1, and y1 = 3 y − y1 = m( x − x1 ) y − 3 = 4( x − 1) y − 3 = 4x − 4 y = 4x −1 Practice Exercise Write the point-slope of the equation of the line passing throuhg the points (3,5) and (8,15). Then solve the equation for y. Answer Point-slope form of the equation: y − 5 = 2( x − 3). Then solve for y gives: y = 2x −1 Example 4: Writing the PointSlope Equation of a Line Write the point-slope form of the equation of the line passing through the points (3,5) and (8,15). Then slove the equation for y. Solution First find the slope to use the point-slope form. Given (3,5) and (8,15). 15 − 5 10 m= = =2 5 8−3 We can take either point on the line to be ( x1 , y1 ). Let's use ( x1 , y1 ) = (3,5). y − 15 = 2( x − 3) y − 15 = 2 x − 6 y = 2x + 9 Practice Exercises 1. Write the point-slope form of the equation of the line passing through (4, −1) with a slope of 8. Then slove the equation for y. 2. Write the point-slope form of the equation of the line passing through the points ( −2, 0) and (0, 2). Then slove the equation for y. Answers to Practice Exercises 1. y = 8 x − 33 2. y = x + 2 The Slope-Intercept Form of the Equation of a Line y The slope-intercept equation of a nonvertical line with slope m and y -intercept b is y = mx + b Y-intercept is b (0, b) Slope is m x A line with slope m and y -intercept b. Graphing y=mx+b Using the Slope and y-Intercept. Plot the y-intercept on the y-axis. This is the point (0,b). Obtain a second point using the slope, m. Write m as a fraction, and use rise over run starting at the y-intercept to plot this point Graphing y=mx+b Using the Slope and y-Intercept. Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of line to show that the line to show that the line continues indefinitely in both directions. Example 5: Graphing by Using the Slope and y-Intercept Give the slope and the y -intercept of the line y = 3 x + 2. Then graph the line. Solution y = 3x + 2 The slope is 3 The y -intercept is 2. 2 Rise Slope = m = 2 = = 1 Run First use the y -intercept 2, to plot the point (0, 2). Starting • • at (0, 2), move 2 units up and 1 unit to the right. This gives us the second point of the line. Use a straightedge to draw a line through the two points. The graph of y = 3 x + 2. Practice Exercises Give the slope and y -intercept of each line whose equation is given. Then graph the line. 1. y = −3 x + 2 3 2. y = x − 3 4 Answers to Practice Exercises 1. m = −3, b = 2 3 2. m = , b = −3 4 Equation of a Horizontal Line A horizontal line is given by an m=0 Y-intercept is 4 • equation of the form y=b where b is the y -intercept. The graph of y = 4 Equation of a Vertical Line A vertical line is given by an X-intercept is -5 equation of the form x = a where a is the x-intercept. • Slope is undefined The graph of x = -5 Example 6: Graphing a Horizontal Line Graph y = 5 in the rectangular coordinate system. Y-intercept is 5. Solution All points on the graph of y = 5 have a value of y that is always 5. Thus it is a horizontal line with y -intercept 5. • Example 7: Graphing a Vertical Line Graph x = −5 in the rectangular coordinate system. Solution No matter what the y -coordinate is, the corresponding x-coordinate for every point on the line is 5. X-intercept is –5. • Practice Exercises Graph each equation in the rectangular coordinate system. 1. y = 4 2. x = 0 Answers to Practice Exercises 1. 2. 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 three real numbers, and A and B are not both zero. Equations of Lines 1. Point-slope form: 2. Slope-intercept form: y − y1 = m( x − x1 ) y = mx + b 3. Horizontal line: y=b 4. Vertical line: x=a 5. General form: Ax + By + C = 0 Example 8: Finding the Slope and the y-Intercept Find the slope and the y -intercept of the line whose equation is 4 x + 6 y + 12 = 0. Solution First rewrite the equation in slope-intercept form y = mx + b. We need to solve for y. 4x + 6 y +12 = 0 6 y = −4 x − 12 The coefficient of x, 4 12 y = − x− 6 6 the constant term, 2, 2 y = − x−2 3 − 23 , is the slope and is the y -intercept. m = − , b = −2. 2 3 Practice Exercises a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. 1. 6 x − 5 y − 20 = 0 2. 4 y + 28 = 0 Answers to Practice Exercises 1. y = x − 4 6 5 6 slope = m = 5 y -intercept = b = −4. Answers to Practice Exercises 2. y = −7 slope = m = 0 y -intercept = b = −7.

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