Section 3.4Slope and Rate of Change

Objectives

Find the slope of a line from its graph Find the slope of a line given two points Find slopes of horizontal and vertical lines Solve applications of slope Calculate rates of change Determine whether lines are parallel or

perpendicular using slope

Objective 1: Find the Slope of a Line from Its Graph The slope of a line is a ratio that compares the vertical

change with the corresponding horizontal change as we move along the line from one point to another.

EXAMPLE 1 Find the slope of the line graphed in the figure

(a) below.

(a)

We can generalize the graphic method for finding slope to develop a slope formula. To begin, we select points P and Q on the line shown in the

figure below. To distinguish between the coordinates of these points, we use subscript notation.

Point P has coordinates (x1, y1), which are read as “x sub 1 and y sub 1.”

Point Q has coordinates (x2, y2), which are read as “x sub 2 and y sub 2.”

Objective 2: Find the Slope of a Line Given Two Points

EXAMPLE 3 Find the slope of the line that passes through

(−2, 4) and (5, −6).

Objective 2: Find the Slope of a Line Given Two Points

In general, lines that rise from left to right have a positive slope. Lines that fall from left to right have a negative slope.

Objective 2: Find the Slope of a Line Given Two Points In the following figure (a), we see a line with slope 3 is steeper than a

line with slope of 5/6, and a line with slope of 5/6 is steeper than a line with slope of 1/4. In general, the larger the absolute value of the slope, the steeper the line.

Lines with slopes of 1 and −1 and are graphed in figure (b) below. When m = 1, the rise and run are, of course, the same number. When m = −1 the rise and run are opposites. Note that both lines create a 45° angle with the horizontal x-axis.

Objective 3: Find Slopes of Horizontal and Vertical Lines

In this section, we calculate the slope of a horizontal line and we show that a vertical line has no defined slope. The y-coordinates of any two points on a horizontal

line will be the same, and the x-coordinates will be different.

Thus, the numerator of y2 − y1 / x2 − x1 will always be zero, and the denominator will always be nonzero. Therefore, the slope of a horizontal line is 0.

EXAMPLE 4

Find the slope of the line y = 3.

Objective 3: Find Slopes of Horizontal and Vertical Lines

The y-coordinates of any two points on a vertical line will be different, and the x-coordinates will be the same.

Thus, the numerator of y2 − y1 / x2 − x1 will always be nonzero, and the denominator will always be 0. Therefore, the slope of a vertical line is undefined.

Objective 4: Solve Applications of Slope The concept of slope has many applications. For example,

Architects use slope when designing ramps and roofs. Truckers must be aware of the slope, or grade, of the roads they travel. Mountain bikers ride up rocky trails and snow skiers speed down steep

slopes.

EXAMPLE 6 Pitch is the incline of a roof expressed as a ratio of the

vertical rise to the horizontal run. Find the pitch of the roof shown in the illustration.

Architecture

Objective 5: Calculate Rates of Change

We have seen that the slope of a line compares the change in y to the change in x. This is calledthe rate of change of y with respect to x.

In our daily lives, we often make many suchcomparisons of the change in one quantity with respect to another.

For example, we might speak of snow melting at the rate of 6 inches per day or a tourist exchanging money at the rate of 12 pesos per dollar.

EXAMPLE 7 A bank offers a business account with a fixed

monthly fee, plus a service charge for each check written. The relationship between the monthly cost y and the number x of checks written is graphed below. At what rate does the monthly cost change?

Banking

Objective 6: Determine Whether Lines Are Parallel or Perpendicular Using Slope Two lines that lie in the same plane but do not intersect are called

parallel lines. Parallel lines have the same slope and different y-intercepts. For

example, the lines graphed in figure (a) are parallel because they both have slope −2/3.

Objective 6: Determine Whether Lines Are Parallel or Perpendicular Using Slope

Slopes of Parallel and Perpendicular Lines: 1. Two lines with the same slope are parallel. 2. Two lines are perpendicular if the product of the

slopes is −1; that is, if their slopes are negative reciprocals.

3. A horizontal line is perpendicular to any vertical line, and vice versa.

EXAMPLE 8

Determine whether the line that passes through (7, −9) and (10, 2) and the line that passes through (0, 1) and (3, 12) are parallel, perpendicular, or neither.