Chapter 1 Linear Functions and Mathematical Modeling

Section 1.2

Section 1.2Slope of a Line, Vertical Intercept,

and Rate of Change

• Concept of Slope

• Slope and the Graph of a Line

• Slope-Intercept Form of a Line

• Slope as Rate of Change

Slope of a LineMeasure of the steepness of a line

Note: Delta (uppercase Δ) is the fourth letter of the Greek alphabet, and it is used frequently to represent “change.” y means "change in y" and x means “

“change in x."

12

12

xxyy

runrise

xy

m

Find the slope of the line through the points (5, –16) and (–7, 8).

or

12

12

xxyy

runrise

xy

m

57)16(8

m

)7(5816

m

1224

1224

2

2

Slopes of Lines (Moving from left to right on the graph)

Slope-Intercept Form of a Line: y = mx + b where m is the slope and b is the y-intercept

Example:

y = –3x – 18

m b

The slope, m = –3.

The y-intercept, b = –18.

Recall that we can find the y-intercept by letting x = 0 and solving for y. Thus, we can verify that b is the y-intercept: y = – 3(0) –18 = –18. So, the y-intercept is (0,–18).

Caution: Before identifying m and b, “y” must be isolated!

True of False: The line 5y = 20x + 30 is written in slope-intercept form.

Slope-intercept form of a line: y = mx + b Answer: False (“y” is not isolated)

Changing to slope-intercept form:

5y = 20x + 305 5 5

y = 4x + 6 where m = 4 (slope) and b = 6 (y-intercept)

Graph the equation by using the slope-intercept method.

Identify the y-intercept and the slope.y-intercept is –2 or (0, –2)

slope is 3/4 Plot the y-intercept, (0, –2).From the y-intercept, “rise” 3 units and “run” 4 units to find a second point. Connect the points with a straight line.

 

2x43

y

Sketch the graph of the line with slope m = –3/5 that passes through the point (–5, 5).

Start by plotting the given point, (–5, 5).

Slope is –3/5: “rise” of –3 and “run” of 5.

From (–5, 5), move down 3 units and right 5 units to find a second point.

Connect the points with a straight line.

 

Slope as Rate of Change

The rate of change describes how one quantity changes in relation to another quantity.

In the context of linear applications, slope is identified as a constant rate of change.

 

Slope as Rate of Change

A helpful way of interpreting slope in the context of a problem is to identify the units on the slope. Write a ratio where the numerator is what the y represents, and the denominator is what the x represents.

Example: A traveler has $4500 in his vacations account and plans to spend $300 per week. The linear equation y = 4500 – 300x represents the amount of money in the account, y, in terms of the number of weeks, x. Interpret the slope.

Slope:

So, we can write

The slope is –300 (equivalent to –300/1); this means that the money decreases at a rate of 300 dollars per week.

 

ΔxΔy

m

.weekdollar

m

A jewelry crafter charges a design fee of $25 for fine earrings and sells each pair for $169. The total cost for the earrings, y, can be represented by the equation y = 169x + 25, where x represents the number of pairs of earrings.

a. Find the slope (rate of change) of the total cost equation, and explain its meaning in the context of this problem.

The line is given in slope-intercept form, where the slope m = 169.

Excluding the design fee, each pair of earrings costs $169.

b. Find the vertical intercept (y-intercept) and explain its meaning.

The vertical intercept is (0, 25).There is an initial design fee of $25.

 

 

1169

earringsco$t

m

The graph below shows the amount of gasoline, g, left in the tank fuel of a 5-gallon portable generator after running for h hours.

a. Find the slope, to the nearest tenth, and explain its meaning.b. Write the equation of this line in slope-intercept form.

a. Using the intercepts, (0, 5) and (8, 0):

The fuel is decreasing at a rate of 0.6 gallons per hour.

b. The slope is –0.6 and the vertical intercept is (0, 5), therefore the equation is g = –0.6h + 5 or g = 5 – 0.6h.

 

 

0.685

0850

ΔhΔg

m hour

gallonsm

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 1.2.