12 4 PointSlope Form Warm Up Problem of

12 -4 Point-Slope Form Warm Up Problem of the Day Lesson Presentation Course 3

12 -4 Point-Slope Form Warm Up Write the equation of the line that passes through each pair of points in slope-intercept form. 1. (0, – 3) and (2, – 3) 2. (5, – 3) and (5, 1) 3. (– 6, 0) and (0, – 2) 4. (4, 6) and (– 2, 0) Course 3 y = – 3 x=5 y = – 1 x – 2 3 y=x+2

12 -4 Point-Slope Form Problem of the Day Without using equations for horizontal or vertical lines, write the equations of four lines that form a square. Possible answer: y = x + 2, y = x – 2, y = –x + 2, y = –x – 2 Course 3

12 -4 Point-Slope Form Learn to find the equation of a line given one point and the slope. Course 3

12 -4 Point-Slope Insert Lesson Title Here Form Vocabulary point-slope form Course 3

12 -4 Point-Slope Form The point-slope form of an equation of a line with slope m passing through (x 1, y 1) is y – y 1 = m(x – x 1). Point on the line (x 1 , y 1 ) Point-slope form y – y 1 = m ( x – x 1) slope Course 3

12 -4 Point-Slope Form Additional Example 1 A: Using Point-Slope Form to Identify Information About a Line Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. y – 7 = 3(x – 4) y – y 1 = m(x – x 1) The equation is in point-slope y – 7 = 3(x – 4) form. Read the value of m from the m=3 equation. (x 1, y 1) = (4, 7) Read the point from the equation. The line defined by y – 7 = 3(x – 4) has slope 3, and passes through the point (4, 7). Course 3

12 -4 Point-Slope Form Additional Example 1 B: Using Point-Slope Form to Identify Information About a Line y– 1=1 3(x + 6) y – y 1 = m(x – x 1) y – 1 = 1 (x + 6) 3 Rewrite using subtraction y – 1 =1 [x – (– 6)] 3 instead of addition. m =1 3 (x 1, y 1) = (– 6, 1) The line defined by y – 1 = 1 (x + 6) has slope 1 , and 3 3 passes through the point (– 6, 1). Course 3

12 -4 Point-Slope Form Check It Out: Example 1 A Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. y – 5 = 2 (x – 2) y – y 1 = m(x – x 1) y – 5 = 2(x – 2) The equation is in point-slope form. Read the value of m from the m=2 equation. (x 1, y 1) = (2, 5) Read the point from the equation. The line defined by y – 5 = 2(x – 2) has slope 2, and passes through the point (2, 5). Course 3

12 -4 Point-Slope Form Check It Out: Example 1 B 2 y – 2 = 3(x + 3) y – y 1 = m(x – x 1) y – 2 = 2 (x + 3) 3 y – 2 = 2 [x – (– 3)] 3 m =2 3 (x 1, y 1) = (– 3, 2) Rewrite using subtraction instead of addition. The line defined by y – 2 = 2 (x + 3) has slope 2 , and 3 3 passes through the point (– 3, 2). Course 3

12 -4 Point-Slope Form Additional Example 2 A: Writing the Point-Slope Form of an Equation Write the point-slope form of the equation with the given slope that passes through the indicated point. the line with slope 4 passing through (5, – 2) y – y 1 = m(x – x 1) [y – (– 2)] = 4(x – 5) y + 2 = 4(x – 5) Substitute 5 for x 1, – 2 for y 1, and 4 for m. The equation of the line with slope 4 that passes through (5, – 2) in point-slope form is y + 2 = 4(x – 5). Course 3

12 -4 Point-Slope Form Additional Example 2 B: Writing the Point-Slope Form of an Equation the line with slope – 5 passing through (– 3, 7) y – y 1 = m(x – x 1) y – 7 = -5[x – (– 3)] Substitute – 3 for x 1, 7 for y 1, and – 5 for m. y – 7 = – 5(x + 3) The equation of the line with slope – 5 that passes through (– 3, 7) in point-slope form is y – 7 = – 5(x + 3). Course 3

12 -4 Point-Slope Form Check It Out: Example 2 A Write the point-slope form of the equation with the given slope that passes through the indicated point. the line with slope 2 passing through (2, – 2) y – y 1 = m(x – x 1) [y – (– 2)] = 2(x – 2) y + 2 = 2(x – 2) Substitute 2 for x 1, – 2 for y 1, and 2 for m. The equation of the line with slope 2 that passes through (2, – 2) in point-slope form is y + 2 = 2(x – 2). Course 3

12 -4 Point-Slope Form Check It Out: Example 2 B the line with slope – 4 passing through (– 2, 5) y – y 1 = m(x – x 1) y – 5 = – 4[x – (– 2)] Substitute – 2 for x 1, 5 for y 1, and – 4 for m. y – 5 = – 4(x + 2) The equation of the line with slope – 4 that passes through (– 2, 5) in point-slope form is y – 5 = – 4(x + 2). Course 3

12 -4 Point-Slope Form Additional Example 3: Entertainment Application A roller coaster starts by ascending 20 feet for every 30 feet it moves forward. The coaster starts at a point 18 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 150 feet forward. Assume that the roller coaster travels in a straight line for the first 150 feet. As x increases by 30, y increases by 20, so the slope 2 of the line is 20 or. The line passes through the 30 3 point (0, 18). Course 3

12 -4 Point-Slope Form Additional Example 3 Continued y – y 1 = m(x – x 1) Substitute 0 for x 1, 18 for y 1, y – 18 = 2 (x – 0) 3 and 2 for m. 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 18 = 2 x. Substitute 150 for x 3 to find the value of y. y – 18 = 2 (150) 3 y – 18 = 100 y = 118 The value of y is 118, so the roller coaster will be at a height of 118 feet after traveling 150 feet forward. Course 3

12 -4 Point-Slope Form Check It Out: Example 3 A roller coaster starts by ascending 15 feet for every 45 feet it moves forward. The coaster starts at a point 15 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 300 feet forward. Assume that the roller coaster travels in a straight line for the first 300 feet. As x increases by 45, y increases by 15, so the slope 1 of the line is 15 or. The line passes through the 45 3 point (0, 15). Course 3

12 -4 Point-Slope Form Check It Out: Example 3 Continued y – y 1 = m(x – x 1) Substitute 0 for x 1, 15 for y 1, y – 15 = 1 (x – 0) 3 and 1 for m. 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 15 = 1 x. Substitute 300 for x 3 to find the value of y. y – 15 = 1 (300) 3 y – 15 = 100 y = 115 The value of y is 115, so the roller coaster will be at a height of 115 feet after traveling 300 feet forward. Course 3

12 -4 Point-Slope Insert Lesson Form Title Here Lesson Quiz Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. 1. y + 6 = 2(x + 5) (– 5, – 6), 2 2. y – 4 = 2 – 5 (x 2 (6, 4), – – 6) 5 Write the point-slope form of the equation with the given slope that passes through the indicated point. 3. the line with slope 4 passing through (3, 5) y – 5 = 4(x – 3) 4. the line with slope – 2 passing through (– 2, 4) y – 4 = – 2(x + 2) Course 3