Linear Functions6.1 SLOPE OF A LINE

Today’s Objectives

• Demonstrate an understanding of slope with respect to: rise and run; rate of change; and line segments and lines, including:• Determine the slope of a line or line segment using

rise and run• Classify a line as having either positive or negative

slope• Explain the slope of a horizontal or vertical line• Explain why the slope can be found using any two

points on the graph of the line or line • Draw a line segment given its slope and a point on the

line

Slope of a Line

• The slope of a line segment is a measure of its steepness• This means a comparison between the vertical change

and the horizontal change:• The vertical change (is called the rise• The horizontal change is called the run

• Slope is normally represented by the lowercase “m”.• We can calculate the slope in several ways such as by

counting or using coordinates of two points on the line• m = slope = =

Slope of a Line A) CountingB) Slope formula

A (-2,1)

B(4,-2)

Down 3

Right 6

Slope = rise/runSlope = -3/6Slope = -1/2

(x1,y1)

(x2,y2)

Slope = rise/run = y2-y1/x2-x1

Slope = [-2-1]/[4-(-2)]Slope = -3/6 = -1/2

Slope of a Line

• If the line segment goes downward from left to right, it will have a negative slope. (rise = negative)• If the line segment goes upwards from left to

right, it will have a positive slope. (rise = positive)

• *The steeper the line goes up or down, the greater the slope.

Horizontal and Vertical Lines

• If a line is horizontal, that is, the rise is equal to zero, then the slope will also be equal to zero.• Slope = = = 0• If a line is vertical, that is, the run is equal to

zero, then the slope of the line will be undefined.• Slope == = undefined

Example

• Find the slopes of the following line segments. Which line segment has the steepest slope? Graph the line segments.

• A) A(-1, 7) B(4, -3)

• B) A(-20, 3) B(-4, -5)

Solutions

Slope of line a) = -10/5 = -2 Slope of line b) = -8/16 = -1/2Line segment in a) is steeper than line segment b)

(-20,3)

(-4,-5)

(-1,7)

(4,-3)

• We can also use the slope formula to find the coordinates of an unknown point on the line when we know the slope and another point on the line.

slope of a line = • Example) Determine the slope of the line that passes

through E(4,-5) and F(8,6)• Solution:Sketch the line. Use the slope of a line formula.

y2 = 6 y1 = -5

x2 = 8 x1 = 4y2 – y1 = 6 – (-5) = 11 = 2.75

• x2 – x1 8 - 4 4

Example) Determine the slope of the following graph:

Known point (5, -6)

(12, -8)

(-2, -4)

y2 – y1 = -8 – (-6) = -2 = -0.29

x2 – x1 12 – (-5) 7

Example) Anna has a part time job. She recorded the hours she worked and her pay for three different days.

Use the table to draw a graph that represents the data.

Slope: 12

The slope represents Anna’s hourly rate of pay.

Anna earned $42 in 3.5 hours.

It took Anna 2.5 hours to earn $30.

Time (h) Pay ($)0 02 244 486 72