Copyright © 2013 Pearson Education, Inc. Section 3.6 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. Point-Slope Form Page 215

Example 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 (x 1, y 1 ) = (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 – y 1 = m(x – x 1 ) 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. Page 215

Example 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. Page 216

Example (cont) We can use either (−2, 3) or (2, 5) for (x 1, y 1 ) 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 x 1 = 2 and y 1 = 5 becomes y – y 1 = m(x – x 1 ) Page 216

Point-Slope FormSOLUTION 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). Substitute the given values Distribute Subtract 5 from both sides Simplify This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form.

Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.5 Point-Slope FormSOLUTION First find the slope of the line. This is done as follows: 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).

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

Example 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 m 1 = 1. The slope of the perpendicular line is m 2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows. Page 218

Example 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. Page 220 similar to Example 7

Example (cont) 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 – y 1 = m(x – x 1 ) y – 8000 = −2000(x – 1) y – 8000 = −2000x y = −2000x + 10,000 A slope of −2000, means that the pump is removing 2000 gallons per hour. Page 220 similar to Example 7

Example (cont) 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. Page 220 similar to Example 7

Example 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). Page 220 similar to Example 7

Example (cont) 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. Page 220 similar to Example 7

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. Figure not in book Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

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. This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form. Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form.

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 Substitute 2008 or 8 for x and compute y. The model predicts that the tuition in 2008 will be $7018 and the tuition in 2015 will be $9090. Use the equation to predict the average cost of tuition and fees at public four-year colleges in Substitute 2015 or15 for x and compute y.

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 Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope. (10, 30.0) (30, 35.3) The slope indicates that each year the median age of the U.S. population is increasing by year.

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 The slope indicates that each year the median age of the U.S. population is increasing by year. A linear equation that models the median age of the U.S. population, y, x years after This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form. (10, 30.0) (30, 35.3)

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 The slope indicates that each year the median age of the U.S. population is increasing by year. A linear equation that models the median age of the U.S. population, y, x years after Use the equation to predict the median age in Because 2020 is 50 years after 1970, substitute 50 for x and compute y. The model predicts that the median age of the U.S. population in 2020 will be (10, 30.0) (30, 35.3)

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 Refer to graph in Figure 3.48 on page 214. a)The slope of the line passing through (1995, 690) and ( ) is b)The number of female officers increased, on average by about 28 officers per year. c)Estimate how many female officers there were in Write the slope-intercept form of the of the line shown in the graph. (1995, 690) (2010, 1110) //

DONE

Group Activity (3.5 on page 214) Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 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.

Objectives Derivation of Point-Slope Form Finding Point-Slope Form Applications