Algebra Textbook Volume 1 Student Edition wc

6.1A (cont’d) Name_______________________________
Consider the following vertical line: y

x

Select four points on this line and list their coordinates:

1) ( , )

2) ( , )

3) ( , )

4) ( , )

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6.1A (cont’d) Name_______________________________

Are there restrictions on the y coordinate of points on the line? ________________________
What is the requirement for the x coordinates? ______________________________________
Thus, the equation for the line indicates this and is x= 3.
Match the equations with the lines they represent:
A. x = 1
B. x = -3
C. x = 6
D. x = -7
E. x = -2

y

x

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6.1A (cont’d) Name_______________________________
Consider the following horizontal line:

y

x

Select four points on this line and list their coordinates:
1) _____________
2) _____________
3) _____________
4) _____________

What do they have in common?

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6.1A (cont’d) Name_______________________________

Are there restrictions on the x-coordinates of points on the line? _________________________
What is the requirement for the y-coordinate? _______________________________________

Thus, the equation for the line indicates this and is y = -2

Match the equations with the line they represent: y

A. y = -2
B. y = 4
C. y = -4
D. y = 1

x

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6.1A (cont’d) Name_______________________________

A common misconception needs to be addressed at this point. Students often draw the
incorrect conclusion that lines with equations in the form x = a number are parallel to the x-axis
(which they are not) and that lines with equations in the form y = a number are parallel to the
y-axis (which they are not). The previous pages explain why this is not the case. Nonetheless, it
is often useful to remember the following rule: Equations of the form x = a number or y = a
number are parallel to the opposite axis. This means lines for x = a number are parallel to the y-
axis and lines for y = a number are parallel to the x-axis.

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6.2A The Slope of a Line Name__________________________

Equations with x and y values define sets of ordered pairs. Some equations, called
__________________equations, give sets of ordered pairs that correspond to straight lines
when graphed.

There are several aspects of a line that can be used to describe and then graph the line. The
three most common are the __________________, the x-intercept, and the y-intercept.

The Slope
The slope of a line is the slant of the line. It can be positive or negative. A line that rises from
left to right has a positive slope. All of the lines below have positive slopes.

y

x

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6.2A cont’d Name_______________________________

A line that falls from left to right has a negative slope. All of the lines below have a negative
slope.

y

x

A horizontal line has no slant. It has a slope of zero.
A vertical line has an undefined slope, which will be explained later in this chapter.
Specifically, the slope of a line is defined as the measure of the rise of the line for an increase of
one on the x-axis. For one step to the right on the x-axis the slope is the change in the y-
coordinate. The following examples will help to illustrate this:

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6.2A (cont’d) Name_______________________________

C B
y

A

x

By examining the lines drawn above, it can be observed that, in the case of line A, for each step

to the right, the line goes up by one (+1). Therefore, its slope is 1 ( 1 1 ). In the case of
ℎ

line B, for each step to the right the line goes up 2. This line, therefore, has a slope of 2

(12ℎ ). The third line (line C) has a slope of ______________. For each step to the right
the line goes up _____ ___________. Draw a line with a slope of 4 on the same graph and label

it D. These slopes are all positive because, as a step to the right is taken, the y-coordinate

increases.

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6.2A (cont’d) Name_______________________________
y

E x

F
G

By examining the lines drawn above, it can be observed that in the case of line E, for each step

to the right, the line goes down by one. Therefore, its slope is -1, which means(11 ). In
ℎ

the case of line F, for each step to the right, the line goes down 2. Therefore, this line has a

slope of -2, which means (21ℎ ). The third line, line G, would have a slope of

_____________ ( ). Draw a line with a slope of -4 on the same graph and label it H.

These slopes are all negative because, as a step to the right is taken, the y-coordinate
decreases.

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6.2A (cont’d) Name_______________________________

There will be some examples where taking only one step to the right will result in a fractional
change in y. To avoid this, select two points on the line with whole number coordinates.

Starting at the point farthest to the left, move right as many spaces as necessary so that moving

up or down will get to the second point. The slope is then (+) (−) . If a
ℎ ℎ

fraction results, it should be reduced if possible.

Example: y

x

3 spaces
up

5 spaces to the right

Slope = 5 3 = 3
ℎ 5

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6.2A (cont’d) Name_______________________________

Another way to view the slope is

ℎ ℎ or
ℎ ℎ

The spaces up or down is the change in y. The spaces to the right mark the change in x. To

compute the rise over the run, simply choose any two points on the line, preferably as close

together as possible. Subtract the x-values of each point from each other and subtract the y-

values from each other.

For example, the last example’s two points were (0, -4) and (5, -1). When subtracting, it doesn’t

matter which coordinate is subtracted as long as you are consistent.
y
−1− −4 = 2− 1 = . Then subtract. The result is the slope.
5−0 2− 1

x −1− −4 = −1+4 = 35.
5− 0 5

The slope of 3 checks out. Slope = 5 3 = 3
5 ℎ 5

Remember the slope of a horizontal line is always _______________ and the slope of a vertical

line is always _______________. Vertical line: Two points on line A are (2,0) and (2,3).

y
A

= 3 − 0 = 3 =
2 − 2 0

x Horizontal line: Two points on line B are (0, −4) and
B (3, −4).

= −4 −(−4) = 0 =
3−0 3

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Practice 6.2B Name_________________________

Determine the slopes of the following lines:

Line A y Line B y

xx

Line C y Line D y

x x

Line E y Line F y

x x

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Practice 6.2C Name_________________________

Determine the slopes of the following lines:

Line A y Line B y

xx

Line C y Line D y

x x

Line E y Line F y

x x

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Practice 6.2D Name_________________________

Determine the slopes of the following lines:

Line A y Line B y

xx

Line C y y
x
Line D

x

Line E y y
x
Line F
x

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Practice 6.2E Name_________________________
Determine the slopes of the following lines:

Line A y Line B y

xx

Line C y Line D y

x x

Line E y Line F y

x x

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6.2E (cont’d) Name_________________________

Determine the slopes of the following lines:

Line G y Line H y

xx

Line I y Line J y

x x

Line K Line L y

y

x x

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6.3A Graphing Using a Point and the Slope Name_________________________

Graph and label each line described below on the coordinate axes provided on the right. The

slope (m) and one point on the line are given. The first one has been done for you.
y

(−4, 7) A.

Examples:

A) Lmin=e −I 3 , (−4,7)
4

B) m = undefined, (−5, 1)

C) m = 3, (−1, −1)

D) m = 0, (3,5) x

E) m = 1 , (4, −3)
4

Line K

Students: y
x
F) m = −1 , (−2,1)

G) m = 0, (2, −5)

H) m = − 4 , (0,6)
3

I) m = 3 , (−6, −4)
2

J) m = −3, (1, −6)

K) m = undefined, (4,7)

L) m = 1 , (−2,1)
2

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6.4A Determining the slope form Two Points (ordered pairs)
Name______________________

As seen earlier, the slope of a line can be determined from the graph of the lines

(# ) . The slope of a line can also be calculated if _____________ points on the
1 ℎ ℎ

line are known. The ___________________in the y-coordinate is divided by the change in the x-

coordinate.

Given: a line through points (x1, y1) and (x2, y2)
The subscripts indicate point #1 and point #2 and it makes no difference which is assigned #1 or
#2.

Formula: y2 – y1 Steps
m = slope = x2 – x1 1) Label the points
2) Use the formula to calculate
Examples: the slope.
Find the slope of the line through the point given: 3) Graph the points and check the
1) (5, 5) and (9, 8) slope of the graph (optional).
x1 y2 x2 y2

y

(9, 8)

+3

(5,5)
+4

Slope (m) = 8−5 = 3
9−5 4

x

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6.4A (cont’d) Name_________________________

Steps: Label the points. Use the formula to calculate the slope. Graph the points and check the
slope on the graph.

x1 y2 x2 y2
2) (-1, -4) and (-3, 8)

m = 8−(−4) = 8+4 = 12 = -6
−3−(−1) −3+1 −2

y

x

Note: When checking the slope, always start with the point on the left.

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6.4A (cont’d) Name_________________________
y

x1 y1 xx y2
3) (3,2) and (-3,2)

m = 2−2 = 0 = 0
−3−3 −6

Note: A calculated slope of (−3, 2) (3, 2)
zero goes along with a
horizontal line which has no
slant.

x

y x1 y1 x2 y2
(2, 3) 4) (2, 3) and (2, -3)

(2, −3) m = −3−3 = −6 = undefined
2−2 0

Remember that division by zero
is “undefined.” Hence the slope
x of vertical lines are referred to as
“undefined.”

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Practice 6.4B Name_________________________
Label the points. Use the formula to calculate the y
slope. Graph the points and check the slope on the
graph. x
1) (0, −2) and (4,4)

2) (2, −1) and (−3, −1) y
3) (6, −3) and (4, −1) x

y
x

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6.4B (cont’d) Name_________________________
4) (−4, −3) and (−4, −1) y

x

5) (1,0) and (3, −6) y
x

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Practice 6.4C Name_________________________

Find the slope of the line connecting the two points. Check with the graph. Two lines can be
graphed on the same set of axes.

y

1) (6,9) and (−4, −11)

x

2) (−3,10) and (12,0)

3) (−3,-5) and (1,7) y
4) (0,8) and (9, −4) x

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6.4C (cont’d) Name_________________________
5) (−5, −1) and (10,8) y

6) (−1,12) and (3, −4) x

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