Rate of ChangeRate of ChangeRate of ChangeRate of Change

Slope, Parallel and Perpendicular Slope, Parallel and Perpendicular LinesLines

When we mention the word slope, most of us think of the slope of a hill or slope of

the roof on a house. There are many representations of the word slope.

Sometimes the vertical change is referred to as the rise, and the horizontal change is referred to as the run. You can remember

slope as rise over run.

slope = rise run

In mathematics, the slope m of a line describes its steepness. The vertical

change is called the change in y, and the horizontal change is called the change in x.

Slopes are either positive, negative, no slope (zero), or undefined. Let’s look at

each.

Positive SlopeLines that have positive slope, slant “up

hill” (as viewed from left to right).Ski Bird has to work hard to make it up the hill. He needs to exert more positive (+)

energy to get up the hill.

Negative SlopeLines that have negative slope, slant “down

hill” (as viewed from left to right).Ski Bird enjoys the ride down the hill. He

needs to decrease (-) energy to try to slow down.

Zero SlopeLines that are horizontal have zero (0)

slope.Ski Bird is cross-country skiing on level

ground. He is not working hard to get up a hill, nor is he trying to slow down. His

energy level is at zero.

No Slope or Slope UndefinedVertical lines have no slope, or undefined

slope.Ski Bird cannot ski vertically. Sheer doom awaits Ski Bird at the bottom of a vertical

hill.

Finding SlopeFinding SlopeFinding SlopeFinding Slope

Rise over runRise over run

The slope of a line can be determined by using the coordinates of any two points on the line.

The change in y can be found by subtracting the y-coordinates. Likewise, the change in x can be

found by subtracting the x-coordinates.

Find the slope of the line graphed to the

left.

In Quadrant III, (lower left), the change in y is

+2, and the corresponding change in x

is +3. Therefore, the slope of the line is 2/3.

Is the slope of the line the same in Quadrant IV?

+2

+3

RIS

E

RUN

Find the slope of the line graphed to the left.

In Quadrant I, the change in y is -2 and the

corresponding change in x is +1. Therefore the slope

of the line is -2/1 or -2. Is the slope of the line the

same in Quadrant IV?

-2

+1

RIS

E

RUN

Find the slope of the line that contains two points (x1,y1), (x2,y2).

A(-2x1, 5y1) and B(4x2, -5y2). Then graph the line.

Slope = difference in y-coordinates difference in x-coordinates

Slope of line AB = -5 – 5 4 – (-2) = -10 or -5 6 3

Slope can be expressed as:

x

yor

run

riseor

xx

yy

12

12

Finding the slope of a line

Determine if the slope of each line is positive,

negative, 0, or undefined.

1) AB2) EG3) HG4) CH

Which lines are parallel?Which lines are perpendicular?

Using the slope formula, find the slope of the line that contains each pair of points.

1) R(9, -2), S(3, -5)

2) T(14, 3), U(-11, 3)

3) V(-1, -2), X(2, -5)

4) B(-6, -4), C(-8, -3)

Find the slope of each line.

Finding Rate of Change Using a

GraphThe graphs of all the ordered pairs (years x1, amount$$ y1) in the example to the right, lie on a line as shown. So the data are linear.

You can use a graph to find a rate of change (slope). Recall that the independent variable is plotted on the x-axis and the dependent variable is plotted on the y-axis.

Finding Rate of Change Using a TableFor the data at the right, is

the rate of change the same for each pair of consecutive days?

What does the rate of change represent?

The rate of change for each pair of days is 15/1. The rate of change is the same for all the data. It costs $15 for each day a computer is rented after the first day>

Number of Days

Rental Charge

1 $60 2 $75 3 $90 4 $105 5 $120 6 $135

Finding Points on a Line

• A line with a slope of 2/3 contains point A(-3,-6).

• Graph A(-3,-6)• Since the slope is 2/3, you

can find another point on the line by counting up 2 units and then right 3 units.

• Draw a line through A(-3,-6) and the new point. The graph should resemble the graph at the right.

Try this

• A line with a slope -2/1 contains point P(0,3).

• Sketch the line.• Find the coordinates of

a second point on the line.

Finding Points on a Linea. Sketch each line.

b. Find a second point on the line.

1) containing A(1, 4); slope ½

2) containing Q(2, 5); slope 3/5

3) containing P(-2, 4) slope -3/4

4) containing S(-3, 1); slope -2/5

You can also analyze the graphs of horizontal and vertical lines. The two

examples below shows why the slope of a horizontal line is 0, and the slope of a

vertical line is undefined.Using the slope formula, pick two points on each

line and find the slope.

Horizontal Lines

Using points (1,4) (x1, x2) and (4, 4) (x2, y2)

4 – 4 = 04 – 1 = 3 0 divided by 3 = 0

The slope of a horizontal line is 0.

(1, 4) (4, 4)

Vertical Lines

Using points (4, -1) (x1, y1) and (4, 2) (x2y2)

2 – (-1) = 34 - 4 = 0 Dividing by 0 is undefined.

Division by zero is undefined. So, the slope of the vertical line is undefined.

(4, -1)

(4, 2)

The following summarizes what you have learned about slope.

Relating Two Lines in Relating Two Lines in the Planethe Plane

Relating Two Lines in Relating Two Lines in the Planethe PlaneParallel LinesParallel Lines

Perpendicular LinesPerpendicular Lines

Mathematics problems often deal with parallel and perpendicular lines. Since

these are such popular lines, it is important that we remember some information about

their slopes.

Two lines in a plane are parallel if they never

intersect. In the diagram at left, the lines are

parallel and have the same slope. You can use slope to determine if two

lines are parallel.All vertical lines are parallel.

If two distinct nonvertical lines are parallel, then they have

equal slopes.If two distinct lines have equal slopes, then they are parallel.

In the graph below, the two purple lines are parallel. Parallel lines - are lines in

the same plane than never intersect.The equations of both lines have the same

slope, 1/2.

Parallel lines have the same slope.

The symbol to indicate parallel line is two vertical bars. || It looks like the number 11.

q || r means line q is parallel to line r and the slope of line q = slope of line r.

The two lines to the left are parallel.

Sometimes the equations must be rewritten to

discover if the slopes are parallel. Are the lines with

equations 2x – 3y = 10 and -3x + 2y =

2 parallel?

Write both equations in slope-intercept form.

2x – 3y = 10-3y = 10 – 2xY = 10 – 2x

-3 -3 y = 2x – 10 3 3Because the slopes, 2/3 and 3/2 are not

equal,

-3x + 2y = 22y = 2 + 3xY = 2 + 3x

2 2 y = 3x + 1 2the lines are not parallel.

Perpendicular lines:(negative reciprocal slopes!)

If two lines in the coordinate plane are not parallel,

then they intersect at a point. If the lines intersect in such a way that they form right angles, then the

lines are perpendicular.

Perpendicular lines have negative reciprocal slopes.

The symbol to indicate perpendicular is an up-side-down capital T.

Line1 is perpendicular to Line2, so the slope of line2 is the negative reciprocal of line1

2121

1

mmll

To find a negative reciprocal of a number, flip the number over (invert) and negate

that value.

5

15

3

1

1

33

4

5

5

4

21

2

2

1

These lines are perpendicular.Their slopes (m) are negative reciprocals.

(Remember y = mx + b.)

64

1

74

xy

xy

In the graph below, the two purple lines are perpendicular. Perpendicular lines -

are lines that intersect to form right angles.

In the equations of the two lines, the product of their slopes (2 · -½) is -1.

The product of two numbers is -1 if one number is the negative reciprocal of the other, such (2 · -½)

SummaryParallel Lines

All vertical lines are parallel.If 2 distinct lines are parallel, then they have equal slopes.If 2 distinct lines have equal slopes, then they are parallel.

Perpendicular LinesEvery vertical line is perpendicular to every horizontal line.

Two lines are perpendicular if the product of their slopes is -1.

Practice with Parallel Practice with Parallel and Perpendicularand Perpendicular

Practice with Parallel Practice with Parallel and Perpendicularand Perpendicular

Problem SolvingProblem Solving

1) Is the equation y = 3x + 2 parallel to 2y + 3x = 3?

2) Find the slope of a line parallel to the line whose equation is 3y + 2x = 6.

3) Find the slope of a line perpendicular to the line whose equation is 3y + 2x = 6.

4) Find the equation of a line parallel to the line whose equation is y = -3x + 5 and passes

through point (0, -5).

5) Find the equation of the line perpendicular to the line whose equation is 2y – 4x = 7 and

passes through point (1, 2).