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Math 152 — Rodriguez The Point-Slope Form of the Equation of a Line Blitzer — 2.5 I. Point-Slope Form A. Linear equations we have seen so far: 1. standard form:Ax +By=C 2. slope-intercept form: 3. horizontal line: 4. vertical line: A, B, and C real numbers y = mx + b, where m slope; y-intercept (0,b) y=# x=# B. Let (x1, y1) 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 − y1 x − x1 The point-slope form of the equation is where m is the slope and (x1, y1) 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 − y1 = m (x = (x − x 1) − ) 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 − y1 = 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= 3) Blitzer — 2.5 − 2 , passing through (−1, 5) 3 2) slope= −3, passing through (9, 5) passing through (4, 2) and (6,−2) 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 − Blitzer — 2.5 line L 5 y = 2x+3 2 3 3x − 4y = 8 Slope line L Slope line M 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. Retail Sales (billions of $) 1) The bar graph shows retail sales, in billions of dollars, of nonfood pet supplies from 2001 to 2005. 10 8 6 4 2 0 7.2 9 8.5 7.9 7.6 (4, 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. 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 Sentenced Inmates (per 100,000 residents) 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. 490 485 480 (4, 486) 475 470 (2, 476) 465 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