Section 3.6 Find and Use Slopes of Lines The _______ of a _____________ line is the ratio of __________ change (_____) to the ____________ change (____) between any two point on a line. slope non-vertical vertical rise horizontal run

+ + Section 3.6 Find and Use Slopes of Lines Key Concept: Slopes of Lines in the Coordinate Plane Slope formula: Given two points and , the formula to find the slope of a line is: Example: Find the slope of the line through points (-1, -2) and (-5, 6) + +

Section 3.6 Find and Use Slopes of Lines Negative slope: Positive slope: Line falls from left to right. Line rises from left to right. Slope results in a negative number. Slope results in a positive number.

Section 3.6 Find and Use Slopes of Lines Zero slope: Undefined slope: Horizontal line with 0 rise. Vertical line with 0 run. Can’t divide by 0

opposite reciprocal slopes. **All vertical lines are parallel. Section 3.6 Find and Use Slopes of Lines Parallel lines: Perpendicular lines: Perpendicular lines have opposite reciprocal slopes. Parallel lines have same slope, but different y-intercepts. **All vertical lines are parallel. +/- and flip

Lines have different slopes, but not opposite reciprocals. Section 3.6 Find and Use Slopes of Lines Coincident lines: Intersecting lines not perpendicular: Coincident lines have same slope and same y-intercepts. Lines have different slopes, but not opposite reciprocals. Lines on top of each other

undefined (vertical line) EXAMPLE 1 Find slopes of lines in a coordinate plane Use rise and run to find the slope of the line a that passes through the points (6, 4) and (8, 2). 2 2 Rise ____ Run ____ Slope _______ Use slope formula to find the slope of line d that passes through the points (6, 4) and (6, 0). Can’t divide by 0 undefined (vertical line)

4 2 GUIDED PRACTICE for Example 1 Use the graph to find the slopes of the other two lines. Find the slope of the line b using run and run from points (6, 4) and (4, 0). 4 2 Rise ____ Run ____ Slope _______ Use slope formula to find the slope of the line c that passes through the points (6, 4) and (0, 4). slope is 0 (horizontal line) 3. A vertical line has a slope that is ___________. undefined

4 4 EXAMPLE 2 Find parallel and perpendicular lines Find the slope of line AB that passes through points A(3, 1) and B(4, 5)? 4 Slope of is _________ Slopes of parallel line are the __________. What is the slope of a line parallel to ? _________ same 4 Slopes of perpendicular lines are ____________________. What is the slope of a line perpendicular to ? _______ opposite reciprocals

+ GUIDED PRACTICE for Example 2 Line g passes through (–1, 3) and (4, 1). Find the slope of a parallel line and a perpendicular line. Explain how you know. y2 – y1 x2 – x1 m = = 1 – 3 4 –(–1) = 5 – 2 Slope line g: + Parallel slope _______ Perpendicular slope_______

EXAMPLE 3 Identifying line relationships Given the graphs of two lines, describe the lines as parallel, perpendicular, coincident, or intersecting, but not perpendicular. b) a) coincident parallel

EXAMPLE 3 Identifying line relationships Given the graphs of two lines, describe the lines as parallel, perpendicular, coincident, or intersecting, but not perpendicular. c) d) intersecting, but not perpendicular perpendicular

EXAMPLE 4 Determining steepness or flatness Compare slopes. The __________ the slope number, the ___________ the line. The __________ the slope number, the __________ the line. A fraction slope between 0 and 1 is ____________. bigger steeper smaller flatter flatter The slope of line a is –3 and the slope line b is . Is line a or line b flatter? __________ line b

Find the missing coordinate EXAMPLE 5 Find the missing coordinate Determine the value of x so that a line through the points at (5, 3) and (x, 4) has a slope of . What does x have to be so the denominator is 2 x = 7

GUIDED PRACTICE for Examples 3, 4 and 5 Coincident lines have the ___________ slope and the _____________ y-intercept. same same Compare parallel lines and coincident lines. Explain. Both have the same slopes, parallel lines have different y-intercepts and coincident lines have same y-intercepts. The slope of line a is –4, slope line b is , and slope of line c is 3. Which line is steeper? ______Which line is flatter? _____ line a line b

GUIDED PRACTICE for Examples 3, 4 and 5 Look at the graph of the two lines. What do you know about the slopes? What do you know about the y-intercepts? The same, both . Different y-intercept points, (0, 1) and (0, -3)

GUIDED PRACTICE for Examples 3, 4 and 5 Determine the value of x so that a line through the points at (x, 4) and (6, 5) has a slope of . What does x have to be so the denominator =2 x = 4