Section 4 3: Slope Introduction We use the term Slope to describe how steep a line is as ou move between an two points on the line. The slope or steepness is a ratio of the vertical change in (rise) compared to the horizontal change in (run) between an two points on the line. The letter m is used to denote the slope we sa that rise run = change in change in Positive Slopes Lines that move up as ou move from a point on the left to a point on the right will have a positive slope. If ou move to the right in the direction the run will be a positive number. If ou then move up in the direction the rise will be a positive number. The ratio of a positive number divided b a positive number will make the slope a positive number. 1. change in change in = 5 7 2. change in change in = 4 6 = 2 3 Up 4 Up 5 Right 6 Right 7 Negative Slopes Lines that move down as ou move from a point on the left to a point on the right will have a negative slope. If ou move to the right in the direction the run will be a positive number. If ou then move down in the direction the rise will be a negative number. The ratio of a negative number divided b a positive number will make the slope a negative number. 3. change in change in = 8 4 = 2 1 4. change in change in = 3 4 Right 4 Right 4 Down 3 Down 8 Math 1 Section 4 3 Page 1 215 Eitel
Slopes of horizontal lines ( ) If the points are all on a horizontal line then there will be a zero rise between the points. The slope will have a zero change in There will be a zero on the top of the fraction m will equal change in change in = a nonzero number = slope Slopes of vertical lines (m is undefined) If the points are all on a vertical line then there will a zero change in the run. The slope will have a zero change in There will be a zero on the bottom of the fraction m will be undefined change in a nonzero number = = undefined slope change in Math 1 Section 4 3 Page 2 215 Eitel
The Slope Formula We use the term Slope to describe how steep a line is as ou move between an two points on the line. The slope or steepness is a ratio of the vertical change (rise) in between these 2 points divided b the horizontal change (run) in between these 2 points. It is common to start with the left most point on the line move towards the right most point on the line to determine the vertical change in the horizontal change (run) in. The slope is a ratio (fraction) with the numerator describing the vertical distance between 2 points the denominator describing the horizontal change between the same 2 points. The two points will be shown as ( 1, 1 ) ( 2, 2 ) where 1 1 are the coordinates of the first point 2 2 are the coordinates of the second point. ( 2, 2 ) 2 1 = the change in (rise) ( 1, 1 ) 2 1 = the change in (run) The vertical change in can be found b subtracting the coordinates 2 1 in that order. The horizontal change in can be found b subtracting the coordinates 2 1 in that order. The ratio of the vertical change (rise) in the line compared to the horizontal change (run) is epressed as the slope. Traditionall the letter m is used to represent the slope. The Slope Formula For an two points ( 1, 1 ) 2, 2 the slope 2 1 2 1 Remember: ( 1 ) means if 1 is negative then ( 1 ) will be positive ( 1 ) means If 1 is negative then ( 1 ) will be positive Math 1 Section 4 3 Page 3 215 Eitel
Slopes of horizontal lines ( ) If two points are on a horizontal line then the will have the same coordinates. The difference 2 ( 1 ) will be If there a zero on the top of the fraction the slope m will equal Slopes of vertical lines (m is undefined) If two points are on a vertical line then the will have the same coordinates. The difference 2 ( 1 ) will be If there a zero on the bottom of the fraction the slope m will be undefined Finding the slope of a line given 2 points 1. Use the slope formula 2 1 2 1 to find the slope of the line with the given points. 2. Reduce each ratio but do not change an improper fraction to a mied number. Eample 1 Eample 2 Eample 3 Find the slope of the Find the slope of the Find the slope of the line through line through line through (1, 5) (3, 8) ( 4, 5) ( 2, 4) (4, 2) (2,1) ( 1, 1 ) ( 2, 2 ) ( 1, 1 ) ( 2, 2 ) ( 1, 1 ) ( 2, 2 ) 8 5 4 ( 5) 1 2 3 1 2 ( 4) 2 4 3 4 + 5 2 2 + 4 = 9 8 2 2 = 4 1 Eample 4 Eample 5 Eample 6 Find the slope of the Find the slope of the Find the slope of the line through line through line through (5, 3) (2, 9) ( 1, 1 ) 2, 2 9 3 2 (5) 6 3 = 2 1 ( 5, 3) ( 2, 3) ( 1, 1 ) ( 2, 2 ) 3 ( 3) 2 ( 5) 3 + 3 2 + 5 = 3 = (2, 1) (2, 3) ( 1, 1 ) 2, 2 3 1 2 2 2 = undefined Math 1 Section 4 3 Page 4 215 Eitel
Finding the slope of a line given the equation of the line Finding the slope -intercept for equations with both an term that have alread been solved for. The most useful form for the equation of a line is the equation that has been solved for. If the equation has been solved for then is alone on one side of the equation the other side will have an term a constant term. The coefficient in front of the term is the slope of the line. We use the letter m for the slope. The constant term is the coordinate of the -intercept. The - intercept is the point where the graph of the line crosses (or intercepts) the ais. We use the letter b for the constant term. The Slope -Intercept form of the equation of a line = m + b When the equation of a line has been solved for then the number in front of the term is called m (the slope) the constant is the coordinate of the -intercept is called b. The slope of the line is m the -intercept is the point (, b) To find the slope -intercept of a linear equation with terms: 1. Get the equation in = m + b form b solving for. 2. The slope will be m (the number in front of the term) 3. The -intercept will be the point (, b) Eample 1 Eample 2 Eample 3 = 3 4 + 2 3 4 - intercept is (,2) = 3 4 3 or 3 1 - intercept is (, 4) = 4 1 1 or 1 - intercept is (, 4) Eample 4 Eample 5 Eample 6 = 1 2 5 = 4 5 = 1 2 - intercept is (, 5) 4 5 - intercept is (,) 1 or 1 1 - intercept is (,) Math 1 Section 4 3 Page 5 215 Eitel
Finding the slope for equations with an term that have not been solved for. Equations where both an terms have not been solved for require that ou solve for (get all alone) first. After the equation has been solved for then the coefficient (number in front of the term) is the slope of the line. Eample 1 Eample 2 Find the slope the - intercept Find the slope the - intercept 3 + 4 = 8 (Solve for ) 5 + 2 = 1 (Solve for ) 3 + 4 = 8 (subtract 3 from both sides) 3 3 5 + 2 = 1 (add 5 to both sides) +5 + 5 4 = 3 + 8 (divide b 4 on both sides) 2 = 5 1 (divide b 2 on both sides) 4 4 = 3 4 + 8 4 = 3 4 + 2 3 4 the - intercept is (,2) 2 2 = 5 2 1 2 = 5 2 5 5 2 the - intercept is (, 5) Eample 3 Eample 4 Find the slope the - intercept 2 5 = 15 (Solve for ) Find the slope the - intercept 4 3 =12 (Solve for ) 2 5 = 15 2 2 (subtract 2 from both sides) 4 3 = 12 (add 4 to both sides) +4 + 4 5 = 2 15 (divide b 5 on both sides) 3 = 4 + 12 (divide b 3 on both sides) 5 5 = 2 5 15 5 = 2 5 + 3 3 3 = 4 3 + 12 3 = 4 3 4 2 5 the - intercept is (,3) 4 3 - intercept is (, 4) Math 1 Section 4 3 Page 6 215 Eitel
Equations of the form = constant Equations with an term but no term represent horizontal lines have a slope of. These lines are horizontal. The cross the ais but not the ais. Eample 1 Eample 2 Eample 3 = 3 = 6 2 = 4 converts to = 6 Equations of the for a constant Equations with an term but no term represent vertical lines have a slope that is undefined. These lines are vertical. The cross the ais but not the ais. Eample 4 Eample 5 Eample 6 = 9 m is undefined = 3 m is undefined + 9 = converts to = 9 m is undefined Math 1 Section 4 3 Page 7 215 Eitel
Parallel Lines Two Parallel Lines Have The Same Slope Two Parallel Lines have the eact same slope. If ou know the slope of one of the lines then the other line has the eact same slope. Eample 7 Line h has a slope of 3 5 Eample 8 Line h has a slope of 2 7 If Line k is parallel to line h then line k has a slope of 3 5 also. If Line k is parallel to line h then line k has a slope of 2 7 also. Eample 9 Line h has a slope of 4 If Line k is parallel to line h then line k has a slope of 4 also. Eample 1 Line h has a slope of 1 If Line k is parallel to line h then line k has a slope of 1 also. Math 1 Section 4 3 Page 8 215 Eitel
Perpendicular Lines Perpendicular lines have slopes that are opposite in sign whose numbers are inverses (negative reciprocals) of each other Eamples of negative reciprocals: Eample 1 Eample 2 Eample 3 Eample 4 2 3 2 1 1 3 2 2 3 3 3 5 5 3 Perpendicular Lines have slopes that are opposite in sign whose numbers are inverses of each other. If ou are given the slope of one of the lines then to find the other lineʼs slope ou change the sign invert (flip) the number given for the first slope. Eample 5 Line h has a slope of 3 5 If Line k is perpendicular to line h then line k has a slope of 5 3 Eample 6 Line h has a slope of 2 7 Eample 7 Line h has a slope of 1 4 If Line k is perpendicular to line h then line k has a slope of 7 2 If Line k is perpendicular to line h then line k has a slope of 4 Eample 8 Line h has a slope of 5 If Line k is perpendicular to line h then line k has a slope of 1 5 Math 1 Section 4 3 Page 9 215 Eitel