Math 152 Rodriguez Blitzer 2.5 The Point-Slope Form of the Equation of a Line I. Point-Slope Form A. Linear equations we have seen so far: 1. standard form: Ax +By=C A, B, and C real numbers 2. slope-intercept form: y = mx + b, where m slope; y-intercept (0,b) 3. horizontal line: y = # 4. vertical line: x = # B. Let (x 1, y 1 ) be a specific (fixed) point on the line and (x,y) be a random (general) point on the line. If we plug these into the slope formula and re-arrange it we get: m = y y 1 x x 1 The point-slope form of the equation is where m is the slope and (x 1, y 1 ) is point on the line. II. Using the Point Slope Form to Write the Equation of a Line A. Given a point and slope: plug info into the equation. Example: slope= 4, passing through (2, 5) y y 1 = m (x x 1 ) y = (x ) You will be asked to write the final answer in slope-intercept form so If you are told to use function notation then:
B. Given two points: find the slope, pick one of the points, plug into equation. Example: passing through (3,8) and (5,4) Steps: 1. Find m = 2. Pick a point: 3. Plug: y y 1 = m (x x 1 ) Again, want final answer in slope-intercept form so you d keep going. C. Instructions you will see for hw: Write the point-slope form of the line s equation satisfying the conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation in function notation. 1) slope= 2 3, passing through ( 1, 5) 2) slope= 3, passing through (9, 5) 3) passing through (4, 2) and (6, 2) Blitzer 2.5 Page 2 of 6
III. Parallel Lines A. Parallel lines: two lines that grow at the same rate and so never intersect. B. Parallel lines have the same slope (or are vertical lines). Example 1: Write an equation for line L in point-slope form and slope-intercept form. y =2x + 3 L (6,4) Example 2: Find the point-slope form and slope-intercept form of the equation for the line passing through ( 3,4) and parallel to the line whose equation is y = 2x+5. IV. Perpendicular Lines A. Perpendicular lines: lines that intersect at a 90 angle. C. Perpendicular lines have slopes whose product is 1. D. The slope of one line is the negative reciprocal of the slope of the other line (or one line is horizontal and the other is vertical). E. Line L and M are perpendicular Slope line L Slope line M line L Slope line L Slope line M 5 y = 2x+3 2 3 3x 4y = 8 Blitzer 2.5 Page 3 of 6
F. Example: Find the point-slope form and slope-intercept form of the equation for the line passing through ( 3,4) and perpendicular to the line whose equation is y = 2x+5. V. Parallel and Perpendicular Lines Examples: Use the given conditions to write an equation for each line in point-slope and slope-intercept form. 1) Passing through (8, 3) and parallel to line whose equation is 3x + 4y = 6 2) Passing through (3, 2) and perpendicular to line whose equation is y = 3x+5 Blitzer 2.5 Page 4 of 6
VI. Applications Same application problems as section 2.4 except now we are not given the y-intercept. 1) The bar graph shows retail sales, in billions of dollars, of nonfood pet supplies from 2001 to 2005. 10 8.5 9 8 Retail Sales (billions of $) 6 4 2 0 7.2 7.6 7.9 (1, 7.6) 0 1 2 3 4 5 Years after 2001 a. Let x represent the number of years since 2001. Let y represent the amount spent on nonfood pet supplies, in billions of dollars. Use the coordinates of the points shown to find the point-slope form of the equation of the line that models the amount of money spent on nonfood pet supplies, y, x years after 2001. (4, 9) b. Write the equation from part (a) in slope-intercept form. c. If this trend continued, use the equation from part (b) to predict the amount spent on nonfood dog supplies in 2010. Blitzer 2.5 Page 5 of 6
2) The following graph is a scatter plot that shows the number of sentenced inmates in the U.S. per 100,000 residents from 2001 to 2005. Also shown is a line that passes through or near the data points. Sentenced Inmates (per 100,000 residents) 490 485 480 475 470 465 (4, 486) (2, 476) 0 1 2 3 4 5 Years after 2000 a) Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation that models the number of inmates per 100,000 residents, y, x years after 2000. b) Write the equation from part (a) in slope-intercept form. Use function notation. c) Use the linear function to predict the number of sentenced inmates in the U.S. per 100,000 residents in 2010. Blitzer 2.5 Page 6 of 6