Slope Created by Charlean Mullikin: mullikinc@anderson3.k12.sc.us ML sections 3.6/3.7

Slope is the relationship of the What is it? Slope is the relationship of the rise to the run of a line. m = rise = y2 – y1 run x2 – x1 rise run

Slope can be positive: + ÷ + or - ÷ - Slope can be negative: + ÷ - or - ÷ +

y's are the same x's are the same Slope can be 0: Horizontal ÷ anything Horizontal x's are the same Slope can be undefined: Anything ÷ Vertical

ALWAYS SIMPLIFY SLOPES Slopes are positive, negative, 0, or Undefined (No slope). Slopes are written as integers with one sign, proper fractions, or improper fractions (no mixed fractions). When 0 is on top, the slope is 0. m = 0 m = -5/-3 When 0 is on bottom, the slope is undefined or no slope. m = undefined m = 1/3 m = 5 m = 5 1/2 m = 5/0 m = 0/6 m = -15/-25 m = 5/2

m = rise = y2 – y1 – run x2 – x1 – (x2 , y2) x2 y2 (x1 , y1) x1 y1 On top!! (x1 , y1) x1 y1 Run On bottom!!

Find the slope of the line that passes through (3, -3)and (0 , 9) m = rise = y2 – y1 = run x2 – x1 – = -12 3 -3 9 – 3 (0 , 9) 9 = -4 Rise On top!! – (3 , -3) 3 -3 – Run On bottom!!

6/2 = 3 -10/-2 = 5 -24/8 = -3 2/6 = 1/3 9/0 = undefined 0/22 = 0

Application Identify rise and run. Which word points to the rise? 3600 feet Put the rise on top. = 16328 feet 3.1 miles What is the run? 3.1 x 5280 = 16368 ft Put the run on bottom. The average slope is about .22. Change to same units, then Divide out and Answer the question in reasonable units.

Slope on a Grid Rise: +9 +6 Run: +6 +9 Slope: +9 +6 m = 3 2

Slope on a Grid Rise: 0 Run: 7 Slope: 0 7 m = 0 +7

Slope on a Grid Rise: -7 Run: +4 Slope: -7 +4 -7 m = 7 4 +4

Slope on a Grid Rise: -8 Run: 0 Slope: -8 -8 m = undefined

YES, Since the slopes are the same (1=1), then the lines ARE PARALLEL. +5 a: m= =1 +5 +5 +2 +2 b: m= =1 +5 +2 +2 YES, Since the slopes are the same (1=1), then the lines ARE PARALLEL.

Perpendicular Lines When two lines are perpendicular, there are two cases with relation to slopes: Case 1-If neither line is vertical, the product of the two slopes is negative one (Opposite reciprocals). m1=2/3 and m2= - 3/2 Case 2 – If one of the lines is vertical, then the perpendicular line is horizontal. m1=undefined and m2= 0

What is the slope of….. 1/2 -2 1/6 -6 3/5 -5/3 -8/7 7/8 No slope 4 Slope of given line Parallel Line? Perpendicular Line? 1/2 -6 3/5 -8/7 4 No slope 1/2 -2 1/6 -6 3/5 -5/3 -8/7 7/8 No slope 4 -1/4 No slope

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Writing Equations

Writing Equations Shortcut #1 Shortcut #2

Writing Equations

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Identify ONE point to use Find slope Substitute

Simplify and solve for y Distributive Property of = Addition Property of = (Add 8 to both sides) Combine like terms Use calculator!

Parallel Equations Lines that are parallel have the same slope. Identify slope of given line Identify point parallel line passes through Use point-slope equation to write equation

Parallel Equations Write the equation of the line parallel to y = ¾ x – 5 that passes through the point (3, -2). m = ¾, parallel slope is also ¾ Point (3, -2) y – y1 = m(x – x1) y - -2 = ¾(x – 3) y + 2 = ¾ x – 9/4 y = ¾ x – 9/4 – 2 y = ¾ x – 17/4

Parallel Equations Write the equation of the line parallel to 7x + 5y = 13 that passes through the point (1, 2). Solve for y to find slope: 7x + 5y = 13 5y = -7x + 13 (subtract 7x from both sides) y = -7/5 x + 13/5 (Divide each term by 5) parallel slope is – 7/5 Point (1, 2) y – y1 = m(x – x1) y - 2 = - 7/5 (x – 1) y - 2 = -7/5 x + 7/5 y = -7/5 x + 7/5 + 2 y = -7/5 x + 17/5

Perpendicular Equations Lines that are perpendicular have slopes that multiply to equal -1. They are opposite sign, reciprocal numbers. Identify slope of given line Change the sign and flip the number to get the perpendicular slope. Use point-slope equation to write equation

Perpendicular Equations Write the equation of the line perpendicular to 7x + 5y = 13 that passes through the point (1, 2). Solve for y to find slope: 7x + 5y = 13 5y = -7x + 13 y = -7/5 x + 13/5 perpendicular slope is +5/7 Point (1, 2) y – y1 = m (x – x1) y - 2 = +5/7(x – 1) y - 2 = 5/7 x – 5/7 y = 5/7 x – 5/7 + 2 y = 5/7 x + 9/7

Perpendicular Equations Write the equation of the line perpendicular to y = ¾ x – 5 that passes through the point (3, -2). m = ¾, perpendicular slope is – 4/3 Point (3, -2) y – y1 = m(x – x1) y - -2 = -4/3(x – 3) y + 2 = -4/3 x + 4 y = -4/3 x + 4 – 2 y = -4/3 x + 2