Chapter 1 Section 4 The Slope of a Straight Line Read pages 24 – 30 Pay particular attention to Examples: 2, 3, 6, 7, and 10

Slope of a Line Recall the general form of the line y = m · x + b Slope is represented by m Geometric Definition of the Slope: m = ( y 2 – y 1 ) / ( x 2 – x 1 )

Exercise 5 (page30) Problem: Find the slope of the line that is formed by the points ( 3, 4 ) and ( 7, 9 ) Solution: Let ( 3, 4 ) be point 1, thus x 1 = 3 and y 1 = 4 Let ( 7, 9 ) be point 2, thus x 2 = 7 and y 2 = 9

Example: Problem 5 (continued) Substituting the values into the equation: m = ( y 2 – y 1 )/( x 2 – x 1 ) = (9 – 4) / ( 7 – 3 ) m = 5/4

Four Properties of the Slope Steepness Property Point – Slope Formula Perpendicular Property Parallel Property

Steepness Property Let L be a line with slope m If we start at any point on the line and move 1 unit to the right, then we must move m units vertically in order to return to the line

Physical Interpretation of the Slope For each increase of 1 “unit (of x) ” in “x”, then “y” increases (if m is positive, or decreases if m is negative) by | m | “units (of y) ”

Exercise 33 part(d) (page 32) x = Number of coats sold y = Total revenue ( in dollars ) y = 100 xm = 100 Interpretation of m = 100 : For each additional coat sold, the total revenue increases by $100.00

Example 2: (not in the book) Let x = Age of House ( in years ) y = Selling Price of House (in $100,000 ) Let y = – x Interpret the meaning of the slope The Slope: m = –

Breakdown of the Interpretation of the Slope “An increase in one unit (of x) of x” means that “the house increases by a year”. “y” is “the selling price of the house” Select “decreases” since the sign of the slope is negative (m = – ) |m| = | – | = “unit (of y) ” is $100,000. So represents $1,334.

Interpretation of the Slope Putting all the information together from the previous slide. Interpretation of the slope: If the age of a house increases by 1 year, then the selling price of the house decreases by $1,334

Point – Slope Formula See Examples 8 and 9 on page 28 The following problem is an alternative to the point slope formula

Exercises 43 (page 32) Problem: Given: ( 5, 0 ) and m = – 7, find the equation on the line. Alternative Solution: Use: y = m · x + b y = ( – 7 ) · x + b ( 0 ) = ( – 7) · ( 5 ) + b b = 35 Thus the equation of the line is: y = – 7 x + 35

Perpendicular Property Technical Definition: –Let m be the slope of line M –Let p be the slope of line P –If line M is perpendicular to line P, then m = – ( 1 / p ) In “Plain English” 1.Take the reciprocal of the slope AND 2.Change the sign

Parallel Property Technical Definition: –Let m be the slope of line M –Let p be the slope of line P –If line M is parallel to line P, then m = p In “Plain English” : The slopes are the same

Example M N P If slope for M is m = ½, then slope for N is n = ½, and the slope of P is p = – 2 (1) M and N are parallel and (2) M and P are perpendicular :