Point-Slope Form 11-4 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

HOMEWORK answers Page 553 #1-8

Pre-Algebra HOMEWORK Page 560 #14-18

Our Learning Goal Students will be able to graph lines using linear equations, understand the slope of a line and graph inequalities.

Our Learning Goal Assignments Learn to identify and graph linear equations. Learn to find the slope of a line and use slope to understand and draw graphs. Learn to use slopes and intercepts to graph linear equations. Learn to find the equation of a line given one point and the slope. Learn to recognize direct variation by graphing tables of data and checking for constant ratios. Learn to graph inequalities on the coordinate plane. Learn to recognize relationships in data and find the equation of a line of best fit.

Today’s Learning Goal Assignment Learn to find the equation of a line given one point and the slope.

Vocabulary point-slope form

The point-slope of an equation of a line with slope m passing through (x1, y1) is y – y1 = m(x – x1). Point on the line Point-slope form y – y1 = m (x – x1) (x1, y1) slope

Additional Example 1: 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. A. y – 7 = 3(x – 4) y – y1 = m(x – x1) The equation is in point-slope form. y – 7 = 3(x – 4) Read the value of m from the equation. m = 3 (x1, y1) = (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).

Additional Example 1B: Using Point-Slope Form to Identify Information About a Line 3 B. y – 1 = (x + 6) y – y1 = m(x – x1) 1 3 y – 1 = (x + 6) y – 1 = [x – (–6)] 1 3 Rewrite using subtraction instead of addition. m = 1 3 (x1, y1) = (–6, 1) The line defined by y – 1 = (x + 6) has slope , and passes through the point (–6, 1). 1 3

Try This: Example 1 Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. A. y – 5 = 2 (x – 2) y – y1 = m(x – x1) The equation is in point-slope form. y – 5 = 2(x – 2) Read the value of m from the equation. m = 2 (x1, y1) = (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).

Try This: Example 1B 2 3 B. y – 2 = (x + 3) y – y1 = m(x – x1) 2 3 y – 2 = (x + 3) y – 2 = [x – (–3)] 2 3 Rewrite using subtraction instead of addition. m = 2 3 (x1, y1) = (–3, 2) The line defined by y – 2 = (x + 3) has slope , and passes through the point (–3, 2). 2 3

Additional Example 2: 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. A. the line with slope 4 passing through (5, -2) y – y1 = m(x – x1) Substitute 5 for x1, –2 for y1, and 4 for m. [y – (–2)] = 4(x – 5) y + 2 = 4(x – 5) The equation of the line with slope 4 that passes through (5, –2) in point-slope form is y + 2 = 4(x – 5).

Try This: Example 2A Write the point-slope form of the equation with the given slope that passes through the indicated point. A. the line with slope 2 passing through (2, –2) y – y1 = m(x – x1) Substitute 2 for x1, –2 for y1, and 2 for m. [y – (–2)] = 2(x – 2) y + 2 = 2(x – 2) The equation of the line with slope 2 that passes through (2, –2) in point-slope form is y + 2 = 2(x – 2).

Additional Example 2: Writing the Point-Slope Form of an Equation B. the line with slope –5 passing through (–3, 7) y – y1 = m(x – x1) Substitute –3 for x1, 7 for y1, and –5 for m. y – 7 = -5[x – (–3)] 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).

Try This: Example 2B B. the line with slope -4 passing through (-2, 5) y – y1 = m(x – x1) Substitute –2 for x1, 5 for y1, and –4 for m. y – 5 = –4[x – (–2)] 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).

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 of the line is or . The line passes through the point (0, 18). 20 30 2 3

Additional Example 3 Continued y – y1 = m(x – x1) Substitute 0 for x1, 18 for y1, and for m. 2 3 y – 18 = (x – 0) 2 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 18 = x. Substitute 150 for x to find the value of y. 2 3 y – 18 = (150) 2 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.

Try This: 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 of the line is or . The line passes through the point (0, 15). 15 45 1 3

Try This: Example 3 Continued y – y1 = m(x – x1) Substitute 0 for x1, 15 for y1, and for m. 1 3 y – 15 = (x – 0) 1 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 15 = x. Substitute 300 for x to find the value of y. 1 3 y – 15 = (300) 1 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.

3. the line with slope 4 passing through (3, 5) 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) 2. y – 4 = – (x – 6) 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) 4. the line with slope –2 passing through (–2, 4) (–5, –6), 2 2 5 (6, 4), – 2 5 y – 5 = 4(x – 3) y – 4 = –2(x + 2)