LinearLinear Equations and Equations and Slope Slope

Created by Laura Ralston Created by Laura Ralston

http://www.youtube.com/watch?v=J_U93-l5Z-w

Slope in Real Life Slope in Real Life

Grade for RoadsGrade for Roads Pitch of Building RoofsPitch of Building Roofs

SlopeSlope

a useful measure of the “steepness” or a useful measure of the “steepness” or “tilt” of a line “tilt” of a line

compares the vertical change (the rise) compares the vertical change (the rise) to the horizontal change (the run) when to the horizontal change (the run) when moving from one point to another along moving from one point to another along the linethe line

typically represented by “m” because it typically represented by “m” because it is the first letter of the French verb, is the first letter of the French verb, monter monter

Formula and Graph Formula and Graph

http://www.youtube.com/watch?v=xBdo-D1RiNs

Four Possibilities of Slope Four Possibilities of Slope

Positive SlopePositive Slope• m > 0m > 0

Line “rises” from Line “rises” from left to rightleft to right

Negative SlopeNegative Slope• m < 0m < 0

Line “falls” from Line “falls” from left to rightleft to right

Four Possibilities of SlopeFour Possibilities of Slope

Zero SlopeZero Slope• m = 0m = 0

Line is horizontal Line is horizontal (constant)(constant)

Undefined SlopeUndefined Slope• m is undefined (0 m is undefined (0

in denominator of in denominator of ratio)ratio)

Line is vertical Line is vertical and is NOT a and is NOT a functionfunction

Do not say “NO Do not say “NO slope”slope”

Linear functions can take on Linear functions can take on many forms many forms

a) Point Slope Forma) Point Slope Form

b) Slope Intercept Formb) Slope Intercept Form

c) General (Standard) Form c) General (Standard) Form

SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM

Most useful Most useful graphing formgraphing form

To convert from To convert from standard to slope standard to slope intercept, given intercept, given equation must be equation must be solved for y. solved for y.

To graph, To graph,

• Identify y-intercept (b) Identify y-intercept (b) and slope (m)and slope (m)

• Plot y-intercept (this is Plot y-intercept (this is now your start point)now your start point)

• Use rise/run concept Use rise/run concept to locate other pointsto locate other points

• Draw line through Draw line through pointspoints

y = mx + b y = mx + b

Where m = slope of the lineWhere m = slope of the line

and and

b = y-intercept b = y-intercept

•ExamplesExamples

POINT-SLOPE FORMPOINT-SLOPE FORM

Most useful Most useful symbolic formsymbolic form

““Write the Write the equation of the equation of the line that meet the line that meet the following following conditions…”conditions…”

To convert from To convert from point-slope to point-slope to slope intercept, slope intercept, distribute “m” distribute “m” and collect like and collect like terms. terms.

y = m(x - xy = m(x - x11) + y) + y11

Where m = slope of the line Where m = slope of the line

and and

(x(x11, y, y11) is any point on the line) is any point on the line

•ExamplesExamples

SPECIAL LINEAR SPECIAL LINEAR RELATIONSHIPSRELATIONSHIPS

PARALLEL : Two or more lines that PARALLEL : Two or more lines that run side by siderun side by side• never intersecting never intersecting • always same distance apart always same distance apart • each line has the same slope each line has the same slope

• mm11 = m = m22

PERPENDICULAR : Two lines that PERPENDICULAR : Two lines that intersect to form 4 right angles intersect to form 4 right angles • Called “negative reciprocals of each Called “negative reciprocals of each

other” (flip and change sign)other” (flip and change sign)• Product of the slopes is equal to -1Product of the slopes is equal to -1

mm11mm22 = -1 = -1