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Section 3.6

Section 3.6. Point-Slope Form. Page 215. Point-Slope Form. The line with slope m passing through the point ( x 1 , y 1 ) is given by y – y 1 = m ( x – x 1 ), or equivalently, y = m ( x – x 1 ) + y 1 the point-slope form of a line. Example. Page 215.

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Section 3.6

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  1. Section 3.6 • Point-Slope Form

  2. Page 215 Point-Slope Form • The line with slope m passing through the point (x1, y1) is given by • y – y1 = m(x – x1), • or equivalently, • y = m(x – x1)+ y1 • the point-slope form of a line.

  3. Example Page 215 • Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? • Solution • Let m = 2 and (x1, y1) = (3,1) in the point-slope form. • To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. y – y1 = m(x – x1) y − 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2 The point (4, 3) lies on the line because it satisfies the point-slope form.

  4. Example Page 216 • Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5). • Solution • Before we can apply the point-slope form, we must find the slope.

  5. Example (cont) Page 216 • We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following. • If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes y – y1= m(x – x1)

  6. Point-Slope Form Write the point-slope form and then the slope-intercept form of the equation of the line with slope 6 that passes through the point (2,-5). SOLUTION Substitute the given values Simplify This is the equation of the line in point-slope form. Distribute Subtract 5 from both sides This is the equation of the line in slope-intercept form.

  7. Point-Slope Form Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (-2,-1) and (-1,-6). SOLUTION First find the slope of the line. This is done as follows: Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.5

  8. Point-Slope Form CONTINUED Use either point provided. Using (-2,-1). Substitute the given values Simplify This is the equation of the line in point-slope form. Distribute Subtract 1 from both sides This is the equation of the line in slope-intercept form.

  9. Example Page 218 • Find the slope-intercept form of the line perpendicular to • y = x – 3, passing through the point (4, 6). • Solution • The line y = x – 3 has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.

  10. Example Page 220similar to Example 7 • A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons. • a. How fast is the pump removing water? • Solution • The pump removes a total of 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour.

  11. Example (cont) Page 220similar to Example 7 • b. Find the slope-intercept form of a line that models the amount of water in the pool. Interpret the slope. • The line passes through the points (1,8000) and (4, 2000), so the slope is y – y1= m(x – x1) A slope of −2000, means that the pump is removing 2000 gallons per hour. y – 8000= −2000(x – 1) y – 8000= −2000x + 2000 y = −2000x + 10,000

  12. Example (cont) Page 220similar to Example 7 • c. Find the y-intercept and the x-intercept. Interpret each. • The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form. • The x-intercept of 5 indicates that the pool is emptied after 5 hours.

  13. Example Page 220similar to Example 7 • d. Sketch the graph of the amount of water in the pool during the first 5 hours. • The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000).

  14. Example (cont) Page 220similar to Example 7 • e. The point (2, 6000) lies on the graph. Explain its meaning. • The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.

  15. Group Activity (3.5 on page 214) Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130, and in 2010 it was $7610. Note that the known value for 2008 is $6530. Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope. Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr. Substitute 5 for 2005, 10 for 2010, and 8 for 2008. Figure not in book

  16. Group Activity (3.5 on page 214) Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation. Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008. This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form. This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form.

  17. Group Activity (3.5 on page 214) Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008. Substitute 2008 or 8 for x and compute y. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015. Substitute 2015 or15 for x and compute y. The model predicts that the tuition in 2008 will be $7018 and the tuition in 2015 will be $9090.

  18. Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. (30, 35.3) (10, 30.0)

  19. Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. This is the equation of the line in point-slope form. (30, 35.3) This is the equation of the line in slope-intercept form. (10, 30.0) A linear equation that models the median age of the U.S. population, y, x years after 1970.

  20. Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. A linear equation that models the median age of the U.S. population, y, x years after 1970. Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y. (30, 35.3) (10, 30.0) The model predicts that the median age of the U.S. population in 2020 will be 40.6.

  21. Example 6 Modeling female officers (page 219) • In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214. • The slope of the line passing through (1995, 690) and (2010.1110) is • The number of female officers increased, on average by about 28 officers per year. • Estimate how many female officers there were in 2006. (2010, 1110) (1995, 690) // Write the slope-intercept form of the of the line shown in the graph.

  22. DONE

  23. Group Activity (3.5 on page 214) Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008. This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form. The model predicts that the tuition in 2008 will be $7018.

  24. Objectives • Derivation of Point-Slope Form • Finding Point-Slope Form • Applications

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