Unit 4 – Analyze and Graph Linear
Equations, Functions and Relations
First Edition
Lesson 1 – Graphing Linear Equations
TOPICS
4.1.1 Rate of Change and Slope
1 Calculate the rate of change or slope of a linear function given information as
sets of ordered pairs, a table, or a graph.
2 Apply the slope formula.
4.1.2 Intercepts of Linear Equations
1 Calculate the intercepts of an equation.
2 Use the intercepts to plot a line.
4.1.3 Graphing Equations in Slope Intercept
1 Give the slope intercept form of a linear equation and define its parts.
2 Graph a line using the slope intercept formula and derive the equation of a line
from its graph.
4.1.4 Point Slope Form and Standard Form of Linear Equations
1 Give the point slope and standard forms of linear equations and define their
parts.
2 Convert point slope and standard form equations into one another.
3 Apply the appropriate linear equation formula to solve problems.
Lesson 2 – Parallel and Perpendicular Lines
TOPICS
4.2.1 Parallel Lines
1 Define parallel lines.
2 Recognize and create parallel lines on graphs and with equations.
4.2.2 Perpendicular Lines
1 Define perpendicular lines.
2 Recognize and create graphs and equations of perpendicular lines.
Some rights reserved
Monterey Institute for Technology and Education 2011
4.1
4.1.1 Rate of Change and Slope
Learning Objective(s)
1 Calculate the rate of change or slope of a linear function given information as sets of
ordered pairs, a table, or a graph.
2 Apply the slope formula.
Introduction
We experience slopes every day. Think about biking down a hill or climbing a set of
stairs. Both the hill and the stairs have a slope. That means that as we travel along
them, we are moving in two directions at the same time—sideways, and either up or
down. In conversation, we use words like gentle or steep to describe the slope of the
ground or an object. Along a gentle slope, most of the movement is horizontal. Along a
steep slope, the vertical movement is greater.
Slope Formula
The mathematical definition of slope is very similar to our everyday one. In math, slope
is the ratio of the vertical and horizontal changes between two points on a surface or a
line. The vertical change between two points is called the rise, and the horizontal
change is called the run. The slope equals the rise divided by the run:
.
This simple equation is called the slope formula.
The slope formula is often shortened to the phrase “rise over run.” Although it sounds
simple, the slope formula is a powerful tool for calculating and comparing the steepness
of landscapes, structures, and lines.
4.2
Self Check A
What is the slope formula?
A) The vertical change divided by the horizontal change between two points on a line.
B) Rise minus run
C) The sideways movement over the up or down movement along a line or surface.
D) gentle or steep
Objective 1
Calculating Slope from a Graph
We can find the slope of a line on a graph by counting off the rise and the run between
two points. If a line rises 4 units for every 1 unit that it runs, the slope is 4 divided by 1, or
4. A large number like this indicates a steep slope: in this case, the slope goes 4 steps
up for every one step sideways.
If a line rises 2 units while it runs 8 units, the slope is 2 divided by 8, or !. A small
number like this represents a gentle slope. This line goes up only 1 step while it travels 4
steps to the side.
The ratio of rise over run describes the slope of all straight lines. This ratio is constant
between any two points along a straight line, which means that the slope of a straight
line is constant, too, no matter where it is measured along the line.
4.3
Slope Formula and Coordinates
It’s easy to find the slope of a line on a graph by measuring the rise and the run. We can
also find the slope of a straight line if we know the coordinates of any two points on that
line. Every point has a set of coordinates: a y-value and an x-value, written as (x, y). The
x value tells us where a point is horizontally. The y value tells us where the point is
vertically.
Consider two random points on a line. We’ll call them point 1 and point 2. Point 1 has
coordinates (x1, y1) and point 2 has coordinates (x2, y2).
The rise is the vertical distance between the two points, which is the difference between
their y-coordinates. That makes the rise y2 ! y1. The run between these two points is the
difference in the x-coordinates, or x2 ! x1. Since slope equals rise over run, the slope of
the line is y2 ! y1 over x2 ! x1. We’ve now got a new way to write the slope formula and
to calculate the value of a slope. Slope is the difference between the y-coordinates
divided by the difference between the x-coordinates.
Mathematicians commonly use the letter m to represent slope. So this is how you will
most often see the slope formula written in algebra:
. At heart, though, this is
still the simple equation of slope = rise over run.
Objective 2
Calculating Slope from Coordinates
Let’s use the slope formula to calculate the slope of a line, using just the coordinates of
two points. The line passes through the points (1, 4) and (-1, 8). Let’s call the point (1, 4)
point 1, and the point (-1, 8) point 2. That means x1 = 1, y1 = 4, x2 = -1, and y2 = 8.
4.4
Example
The slope formula is
the equation
Points
Coordinates
(1, 4)
x1 = 1
y1 = 4
(-1, 8)
x2 = -1
y2 = 8
. When we put the coordinates into the formula, we get
. This reduces to m = -2. The slope of the line is -2.
It is important to realize that it doesn’t matter which point is designated as 1 and which is
2. We could have called (-1, 8) point 1, and (1, 4) point 2. In that case, putting the
coordinates into the slope formula produces the equation
simplifies to m = -2. That’s the same slope as before.
4.5
. Once again, that
Self Check B
What is the slope of the line between the points (-2, 1) and (1, 3)?
A) 2/3
B) -2
C) -2/3
D) 3/2
Slope of Horizontal and Vertical Lines
So far we’ve considered lines that run “uphill” or “downhill.” Their slopes may be large or
small, but they are always positive or negative numbers. But there are two other kinds of
lines, horizontal and vertical. What is the slope of a flat line or level ground? Of a sheer
wall or a vertical line?
Let’s consider a horizontal line on a graph. No matter which two points we choose on the
line, they will always have the same y-coordinate.
That means the rise, the vertical difference between two points, will always be zero. That
makes sense—a horizontal line doesn’t go up or down. What happens when we put a
rise of 0 into the slope formula? It becomes
. Zero divided by any number is
zero. The slope of a horizontal line is always 0.
How about vertical lines? In their case, no matter which two points we choose, they will
always have the same x-coordinate,
4.6
That means that the run, the horizontal difference between two points, will always be
zero. That makes sense—a vertical line doesn’t go sideways at all. When we put a run of
0 into the slope formula, the equation becomes
. We can’t calculate that
value because division by zero has no meaning in the set of real numbers. All vertical
lines have a slope that is undefined.
Summary
As you can see, slopes play an important role in our everyday life. You may walk up a
slope to get to the bus stop or ski down the slope of a mountain. The slope formula,
written
, is a useful tool you can use to calculate the vertical and horizontal
change of a variety of slopes.
4.7
4.1.1 Self Check Solutions
Self Check A
What is the slope formula?
A) The vertical change divided by the horizontal change between two points on a line.
B) Rise minus run
C) The sideways movement over the up or down movement along a line or surface.
D) gentle or steep
A) Correct. The vertical change divided by the horizontal change between two points on
a line.
B) Incorrect. The correct answer is the vertical change divided by the horizontal change
between two points on a line.
C) Incorrect. That is the opposite of the slope formula. The correct answer is the vertical
change divided by the horizontal change between two points on a line.
D) Incorrect. The terms gentle or steep describe a slope verbally, not mathematically.
The correct answer is the vertical change divided by the horizontal change between two
points on a line.
Self Check B
What is the slope of the line between the points (-2, 1) and (1, 3)?
A) 2/3
B) -2
C) -2/3
D) 3/2
A) Correct.
B) Incorrect. The denominator is 1 ! (-2), not 1 ! 2. The correct answer is
.
C) Incorrect. You did not put the coordinates into the slope formula consistently. The
answer is
.
D) Incorrect. You inverted the rise and the run. The correct answer is
4.8
.
4.1.2 Intercepts of Linear Equations
Learning Objective(s)
1 Calculate the intercepts of a line.
2 Use the intercepts to plot a line.
Introduction
The intercepts of a line are the points where the line intercepts, or crosses, the
horizontal and vertical axes.
The straight line on the graph below intercepts the two coordinate axes. The point where
the line crosses the x-axis is called the x-intercept. The y-intercept is the point where
the line crosses the y-axis.
Notice that the y-intercept occurs where x = 0, and the x-intercept occurs where y = 0.
Objective 1
Calculating Intercepts
We can use the characteristics of intercepts to quickly calculate them from the equation
of a line. Just see how easy it is, as we find the x- and y-intercepts for the line
.
To find the y-intercept, we substitute 0 for x in the equation, because we know that every
point on the y-axis has an x-coordinate of 0. Once we do that, we can solve to find the
4.9
value of y. When we make x = 0, the equation becomes
, which works out
to y = 2. So when x = 0, y = 2. The coordinates of the y-intercept are (0, 2).
Example
Problem
3y + 2x
=
6
3y + 2(0)
=
6
3y
=
6
=
Answer
y
=
2
Now we’ll follow the same steps to find the x-intercept. We’ll let y = 0 in the equation,
and solve for x. When y = 0, the equation for the line becomes
, and that
works out to x = 3. When y = 0, x = 3. The coordinates of the x-intercept are (3, 0).
Example
Problem
3y + 2x
=
6
3(0) + 2x
=
6
2x
=
6
=
Answer
x
=
3
See, I told you that it would be easy.
Self Check A
What is the y-intercept of a line with the equation
A)
B) (-4, 0)
C) (0, -4)
D) (5, -4)
4.10
?
Objective 2
Using Intercepts to Graph Lines
Knowing the intercepts of a line is a useful thing. For one thing, it makes it easy to draw
the graph of a line—we just have to plot the intercepts and then draw a line through
them. Let’s do it with the equation
. We figured out that the intercepts of the
line this equation represents are (0, 2) and (3, 0). That’s all we need to know:
And there we have the line.
Intercepts and Problem-Solving
Intercepts are also valuable tools for predicting or tracking a process. At each intercept,
one of the two quantities being plotted reaches zero. That means that the intercepts of a
line can be used to mark the beginning and the end of a process.
Imagine a student named Morgan who is buying a laptop for $1,080 to use for school.
Morgan is going to use the computer store’s finance plan to make this purchase—she’ll
pay $45 per month for 24 months.
She wants to know how much she will still owe after each month of the plan. She can
keep track of her debt by making a graph. The x-axis will be the number of months and
the y-axis will represent the amount of money she still owes on the finance plan. Morgan
knows two points in her pay-off schedule. The day she buys the computer, she’ll be at 0
months passed and $1,080 owed. The day she pays it off completely, she’ll be at 24
months passed and $0 owed. With these two points, she can draw a line, running from
the y-intercept at (0, 1080) to the x-intercept at (24, 0).
4.11
Morgan can now use this graph to figure out how much money she still owes after any
number of months.
Let’s look at another situation involving intercepts, this time when we know only one
intercept and want to find the other. Joe is a lifeguard at the local swimming pool. It’s the
end of the summer, and the pool is being drained. Joe has to wait by the pool until it’s
completely empty, so no one falls in and drowns. How can poor Joe figure out how long
that’s going to take?
If Joe has taken an algebra course, he’s got it made. The pool contains 10,200 gallons
of water. It drains at a rate of 680 gallons per hour. Joe can use that information to make
a table of how much water will be left in the pool hour by hour.
x, Time
(hours)
y, Volume of Water
(gallons)
0
10,200
1
9,520
2
8,840
3
8,160
4
7,480
Once he’s calculated a few data points, Joe can use a graph and intercepts as a shortcut to find out how long it will be until the pool is dry. Joe’s starting point is the y-
4.12
intercept, where the pool is full at 10,200 gallons and the elapsed time is 0. Next, he
plots the volume of the pool at 1, 2, 3, and finally 4 hours.
Now all Joe needs to do is connect the points with a line, and then extend the line until it
meets the x-axis.
The line intercepts the x-axis when x = 15. So now Joe knows—the pool will take 15
hours to drain completely. It’s going to be a long day.
Summary
We’ve now seen the usefulness of the intercepts of a line. When we know where a line
crosses the x- and y-axes, we can easily produce the graph or the equation for that line.
When we know one of the intercepts and the slope of a line, we can find the beginning or
predict the end of a process.
4.13
4.1.2 Self Check Solutions
Self Check A
What is the y-intercept of a line with the equation
?
A)
B) (-4, 0)
C) (0, -4)
D) (5, -4)
A)
Incorrect.
is the x-intercept. At the y-intercept, x = 0. When 0 is substituted for x in
the equation, y = -4. The correct answer is (0, -4).
B) (-4, 0)
Incorrect. This answer switches the values of x and y. Coordinates are given in the order
(x, y). At the y-intercept, x = 0. When 0 is substituted for x in the equation, y = -4. The
correct answer is (0, -4).
C) (0, -4)
Correct. At the y-intercept, x = 0. When 0 is substituted for x in the equation, y = -4.
D) (5, -4)
Incorrect. This is the co-efficient of x and the constant, not the y-intercept. At the yintercept, x = 0. When 0 is substituted for x in the equation, y = -4. The correct answer is
(0, -4).
4.14
4.1.3 Graphing Equations in Slope Intercept Form
Learning Objective(s)
1 Give the slope intercept form of a linear equation and define its parts.
2 Graph a line using the slope intercept formula and derive the equation of a line from
its graph.
Introduction
Straight lines are produced by linear functions. That means that a straight line can be
described by an equation that takes the form of the linear equation formula,
.
In the formula, y is a dependent variable, x is an independent variable, m is a
constant rate of change, and b is an adjustment that moves the function away from the
origin. In a more general straight line equation, x and y are coordinates, m is the slope,
and b is the y-intercept. Because this equation describes a line in terms of its slope and
its y-intercept, this equation is called the slope-intercept form.
Objective 1
Slope Intercept Formula
Given a linear equation in the form
, if we change m we change the slope, or
steepness, of the line. As m, the slope, gets larger, the line gets steeper. When m gets
smaller, the slope flattens. If we change b, we change the y-intercept. A positive yintercept means the line crosses the y-axis above the origin, while a negative y-intercept
means that the line crosses below the origin.
Simply by changing the values of m and b, we can define any straight line. That’s how
powerful and versatile the slope intercept formula is.
Self Check A
How is the x-intercept represented in the slope intercept form of a linear equation?
A) It is represented by x.
B) It is represented by m.
C) It is represented by b.
D) It is not represented.
Objective 2
From Graph to Equation
Now that we understand the slope intercept form, we can look at the graph of a line and
write its equation just from identifying the slope and the y-intercept. Let’s try it with this
line:
4.15
The slope intercept form is
. For this line, the slope is
, and the y-intercept
is 4. If we put those values into the formula, we get the equation
. That’s the
slope intercept equation of our line.
Self Check B
What is the equation of the line in the graph below?
4.16
A)
B)
C)
D)
From Equation to Graph
We’ve seen that it’s not difficult to convert the graph of a line to an equation. We can
also work the other way and produce a graph from a slope intercept equation. Consider
the equation
. This equation tells us that the y-intercept is at -1. We’ll start by
plotting that point, (0, -1), on a graph.
The equation also tells us that the slope of this line is -3. So we’ll count up 3 units and
over -1 unit and plot a second point. (We could also have gone down 3 and over +1.)
Then we draw a line through both points, and there it is, the graph of
.
4.17
Self Check C
Which graph shows the line
?
A)
B)
C)
D)
Summary
The slope intercept form of a linear equation is written as
, where m is the
slope and b is the value of y at the y-intercept. Because we only need to know the slope
and the y-intercept to write this formula, it is fairly easy to derive the equation of a line
from a graph and to draw the graph of a line from an equation.
4.18
4.1.3 Self Check Solutions
Self Check A
How is the x-intercept represented in the slope intercept form of a linear equation?
A) It is represented by x.
B) It is represented by m.
C) It is represented by b.
D) It is not represented.
A) It is represented by x.
Incorrect. x is an x-coordinate, but not the x-intercept. The slope intercept form of a
linear equation is based on the slope and the y-coordinate at the y-intercept. The correct
answer is that it is not represented.
B) It is represented by m.
Incorrect. m is the slope of the line. The slope intercept form of a linear equation is
based on the slope and the y-coordinate at the y-intercept. The correct answer is that it
is not represented.
C) It is represented by b.
Incorrect. b is the y-coordinate at the y-intercept. The slope intercept form of a linear
equation is based on the slope and the y-coordinate at the y-intercept. The correct
answer is that it is not represented.
D) It is not represented.
Correct. The slope intercept form of a linear equation is based on the slope and the ycoordinate at the y-intercept.
Self Check B
What is the equation of the line in the graph below?
4.19
A)
B)
C)
D)
A)
Incorrect. You have inverted the slope. The correct slope of this line is
intercept is -3. The correct answer is
and the y-
.
B)
Correct. The slope of this line is
and the y-intercept is -3.
C)
Incorrect. 4 is the x-intercept, not the slope. The slope of this line is
intercept is -3. The correct answer is
and the y-
.
D)
Incorrect. The slope is positive and the y-intercept is negative, not the other way around.
The correct answer is
.
Self Check C
4.20
Which graph shows the line
?
A)
B)
C)
D)
A) Graph A
Correct. This line has a positive y-intercept and a steep positive slope, as the equation
requires.
B) Graph B
Incorrect. This line has a gentle slope, while the equation specifies a steep slope. The
correct answer is Graph A.
C) Graph C
Incorrect. This line has a negative y-intercept and a negative slope, while the equation
specifies a positive y-intercept and a steep positive slope. The correct answer is Graph
A.
D) Graph D
Incorrect. This line has a negative y-intercept and a gentle slope, while the equation
specifies a positive y-intercept and a steep positive slope. The correct answer is Graph
A.
4.21
4.1.4 Point Slope Form and Standard Form of Linear Equations
Learning Objective(s)
1 Give the point slope and standard forms of linear equations and define their parts.
2 Convert point slope and standard form equations into one another.
3 Apply the appropriate linear equation formula to solve problems.
Introduction
Linear equations can take several forms, such as the point-slope formula, the slopeintercept formula, and the standard form of a linear equation. These forms allow
mathematicians to describe the exact same line in different ways.
This can be confusing, but it’s actually quite useful. Consider how many different ways
you could write a request for milk on a shopping list. You could ask for white milk, cow’s
milk, a quart of milk, or skim milk, and each of these phrases would describe the exact
same product. The description you use will depend on the characteristics that matter
most to you.
Equations for describing lines can be chosen the same way—they can be written and
manipulated based upon on which characteristics of the line are of interest. Even better,
when a different characteristic becomes important, linear equations can be converted
from one form to another.
Objective 1
Point Slope Form
One type of linear equation is the point slope form, which gives the slope of a line and
the coordinates of a point on it. The point slope form of a linear equation is written as
. In this equation, m is the slope and (x1, y1) are the coordinates of a
point.
Let’s look at where this point-slope formula comes from. Here’s the graph of a generic
line with two points plotted on it.
4.22
The slope of the line is “rise over run.” That’s the vertical change between the two points
(the difference in the y-coordinates) divided by the horizontal change over the same
segment (the difference in the x-coordinates). This can be written as
. This
equation is the slope formula.
Now let’s say that one of these points is a generic point (x, y), which just means it could
be anywhere on the line, and the other point is a specific point,
. If we plug these
coordinates into the formula, we get
. Now we can rearrange the equation a
little bit by multiplying both sides of the formula by
. This simplifies to
.
is the point-slope formula. We’ve converted the slope formula into
the point slope formula. We didn’t do that just for fun, but because the point slope
formula is sometimes more useful than the slope formula, for example when we need to
find the equation of a line when given a point and the slope.
Let’s do an example. Consider a line that passes through the point (1, 3) and has a
slope of
.
4.23
Putting these values into the point-slope formula, we get
. That’s the
equation of the line.
Self Check A
Which of the following points lies on the line (y + 8) = 7(x ! 5)?
A) (5, -8)
B) (5, 8)
C) (8, 5)
D) (8, -5)
Objective 2
Standard Form
Remember, the point-slope formula is only one type of linear equation. It is effective in
describing some of the characteristics of a straight line. However, point slope equations
can be awkward to use in some algebraic operations. In such cases, it may be helpful to
convert the equation into a different form, the standard form.
The standard form of an equation is Ax + By = C. In this kind of equation, x and y are
variables and A, B, and C are integers.
4.24
We can convert a point slope equation into standard form by moving the variables to the
left side of the equation. Let’s go back to that point-slope equation of
.
We can rearrange the terms as follows:
Example
Problem
(y ! 3)
=
4(y ! 3)
=
4y ! 12
=
-1x + 1
x + 4y ! 12
= -x + 1 + x
X + 4y ! 12
= 1
x + 4y ! 12 + 12
Standard Form
x + 4y
= 1 + 12
= 13
When we shift the variable terms to the left side of the equation and everything else to
the right side, we get
. This equation is now in standard form.
Matching the Formula to the Situation
We now know how to convert equations from point slope to standard form, and how to
go back and forth between a graph and a linear equation. But with so many choices,
how do we decide which form to use in a real-life situation?
The answer is to identify what you know and what you want to find out, and see which
form uses those terms. Let’s look at a situation where one form of an equation is more
useful than the others.
Andre wants to buy an MP3 player. He got $50 for his birthday, but the player he wants
costs $230, so he’s going to have to save up the rest. His plan is to save $30 a month
until he has the money he needs. We’ll help him out by writing an equation to analyze
this situation. This will help us to figure out when he will have saved up enough to buy
the MP3 player.
When we write the equation, we’ll let x be the time in months, and y be the amount of
money saved. After 1 month, Andre has $80. That means when x = 1, y = 80. So we
know the line passes through the point (1, 80). Also, we know that Andre hopes to save
$30 per month. This means the rate of change, or slope, is 30.
We have a point and we have a slope—that’s all we need to write a point slope formula,
so that’s the form of linear equation we’ll use. Remember, the point slope form is
4.25
Objective 3
. When we substitute in Andre’s point and slope, the equation
becomes
.
Okay, now what? Well, we have a formula that describes Andre’s savings plan. We can
use that to figure out how long it will take him to save all the money he needs to buy the
MP3 player.
Remember, the y in this equation represents the amount Andre has saved, and the x
represents the number of months he has been saving. We want to find what the value of
x is when y equals 230. So we just need to set y equal to 230 in our equation, and solve
for x.
Example
Problem
y ! 80
=
30(x ! 1)
230 ! 80
=
30(x ! 1)
150
=
30x ! 30
180
=
30x
6
=
x
Answer
The result is x = 6. It will take Andre 6 months to save the $230 he needs to buy the
MP3 player. Because the problem told us that we knew a point and a slope, we were
able to choose the right form for the job of writing an equation. Once we wrote the
equation, we were able to solve it for the variable we wanted to find.
Summary
We’ve learned that linear equations can be written in different forms, depending upon
what we either know or want to know about a line. The point slope form,
, is useful in situations involving slope and the location of one or more
points. The standard form, Ax + By = C, is usually easier to use when we need to make
algebraic calculations. When needs or knowledge change, we can convert an equation
from one form into another.
4.26
4.1.4 Self Check Solutions
Self Check A
Which of the following points lies on the line (y + 8) = 7(x ! 5)?
A) (5, -8)
B) (5, 8)
C) (8, 5)
D) (8, -5)
A) (5, -8)
Correct. The point slope formula is
. In the given equation, x1 is 5 and
y1 is -8. That means that the correct answer is (5, -8).
B) (5, 8)
Incorrect. You’ve missed a sign. The point slope formula is
. In the
given equation, x1 is 5 and y1 is -8. That means that the correct answer is (5, -8).
C) (8, 5)
Incorrect. You’ve reversed x and y and missed a sign. The point slope formula is
. In the given equation, x1 is 5 and y1 is -8. That means that the
correct answer is (5, -8).
D) (-8, 5)
Incorrect. You’ve reversed x and y. The point slope formula is
. In the
given equation, x1 is 5 and y1 is -8. That means that the correct answer is (5, -8).
4.27
4.2.1 Parallel Lines
Learning Objective(s)
1 Define parallel lines.
2 Recognize and create parallel lines on graphs and with equations.
Objective 1
Introduction
Parallel lines are two or more lines that never intersect. Examples of parallel lines are
all around us, in the two sides of this page and in the shelves of a bookcase. When you
see lines or structures that seem to run in the same direction, never cross one another,
and are always the same distance apart, there’s a good chance that they are parallel.
In algebra, we use something more precise than appearance to recognize and create
parallel lines. We use equations.
Objective 2
Recognizing Parallel Lines
Let’s look at two parallel lines on a graph.
Line A has an equation of
. Line B has an equation of
identify what these two lines have in common?
4.28
. Can you
Try comparing the coefficients of x in each equation. They are both 3. Because the
equations are written in slope-intercept form of
, the coefficient of x is the
slope of the lines. Since both Line A and B have a slope of 3, they have the same slope.
We’ve discovered a relationship that is true for all parallel lines—lines are parallel if they
have the same slope. Not convinced? Look at the lines on this graph:
The blue lines are parallel—they run in the same direction, keep the same distance
apart, and never touch—and they have the same slope. All three red lines share a slope,
and they too are parallel to one another. The red and blue lines are clearly not parallel,
and they have different slopes.
Creating Parallel Lines
To create a parallel line from an equation, we start by identifying the slope of the original
line. Then we write a second equation, repeating the slope but changing the y-intercept.
Once again, because there are an infinite number of possible y-intercepts, there are an
infinite number of equations we could create.
4.29
Self Check A
A line has the equation
. Which of the following lines is parallel to it?
A)
B)
C)
D)
Summary
Parallel lines are lines that have the same slope. These lines never intersect and always
maintain the same distance apart. To create a parallel line from an equation, we simply
change the y-intercept.
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4.2.1 Self Check Solutions
Self Check A
A line has the equation
. Which of the following lines is parallel to it?
A)
B)
C)
D)
A)
Incorrect. This line has a different slope than the given line, while parallel lines have the
same slope. The correct answer is
.
B)
Correct. Parallel lines have the same slope.
C)
Incorrect. This is the same line as the one given in the original equation. Parallel lines
have the same slope, but are not the same line. The correct answer is
.
D)
Incorrect. This line has the opposite slope of the given line, and parallel lines have the
exact same slope. The correct answer is
.
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4.2.2 Perpendicular Lines
Learning Objective(s)
1 Define perpendicular lines.
2 Recognize and create graphs and equations of perpendicular lines.
Objective 1
Introduction
Perpendicular lines are two or more lines that intersect at a 90-degree angle, like the
two lines drawn on this graph, and the x and y axes that orient them.
Perpendicular lines are everywhere, not just on graph paper but also in the world around
us, from the crossing pattern of roads at an intersection to the colored lines of a plaid
shirt. In our daily lives, we may be happy to call two lines perpendicular if they merely
seem to be at right angles to one another. In algebra, we use equations to make sure of
it.
Objective 2
Recognizing Perpendicular Lines
Let’s compare the equations of lines that are perpendicular and those that are not to see
if there is a pattern that will allow us to recognize perpendicular lines. We’ll start with two
lines that we know are not perpendicular.
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Line A has an equation of
, and Line B has an equation of
. These
two lines are clearly not at 90-degrees to one another. Do you see any connection
between their equations?
If you couldn’t find anything—you’re right. There isn’t any link between the slopes or yintercepts of these lines.
Now let’s look at two lines that are perpendicular:
4.33
On this graph, Line A has the equation of
and Line B has the equation of
. Do you see any connection between these equations?
If you said no again this time, you missed something. It’s a little tricky, but there is a
relationship between the slopes of these two lines. The slope of Line A is
slope of Line B is
, and the
. These numbers are opposite reciprocals. An opposite reciprocal
is the fraction form of a number flipped upside down (that’s the reciprocal part), and with
its sign changed (that’s the opposite part). Write those slopes side by side and it
becomes more obvious:
and
are opposite reciprocals.
This is the defining feature of the equations of perpendicular lines—the slopes of
perpendicular lines are opposite reciprocals.
4.34
Self Check A
If a line has a slope of
, what is the slope of a line perpendicular to it?
A)
B)
C)
D)
Creating Perpendicular Lines
To create a perpendicular line from an equation, we once again start by identifying the
slope of the original line. Then we write a second equation, changing the slope to the
opposite reciprocal of the slope in the original equation.
Self Check B
Which of the following lines are perpendicular to the line
A)
B)
and
and
C)
D) all of the lines are perpendicular
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?
Summary
Perpendicular lines are lines that cross one another at a 90° angle. They have slopes
that are opposite reciprocals of one another. Unlike parallel lines that never touch,
perpendicular lines must intersect.
4.2.2 Self Check Solutions
Self Check A
If a line has a slope of
, what is the slope of a line perpendicular to it?
A)
B)
C)
D)
A)
Incorrect. This is the same slope as the original line. Perpendicular lines have opposite
reciprocal slopes. The correct answer is
.
B)
Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the opposite
but not the opposite reciprocal. The correct answer is
.
C)
Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the
reciprocal but not the opposite reciprocal. The correct answer is
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.
D)
Correct. Perpendicular lines have opposite reciprocal slopes.
reciprocal of
is the opposite
.
Self Check B
Which of the following lines are perpendicular to the line
A)
and
B)
and
C)
D) all of the lines are perpendicular
A)
and
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?
Correct. These lines both have a slope of
, which I the opposite reciprocal of the
slope of 7 in the original equation. Both these lines are perpendicular to the original line.
B)
and
Incorrect. Perpendicular lines have opposite reciprocal slopes. Both these lines have
reciprocal slopes to the slope of 7 in the original equation, but only the first line is the
opposite reciprocal. The correct answer is
and
.
C)
Incorrect. Perpendicular lines have opposite reciprocal slopes. This slope is the opposite
of the original slope of 7, but not the opposite reciprocal. The correct answer is
and
.
D) all of the lines are perpendicular
Incorrect. Perpendicular lines have opposite reciprocal slopes. The original line has a
slope of 7, so all perpendicular lines must have a slope of
and
.
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. The correct answer is
Unit Recap
4.1.1 Rate of Change and Slope
As you can see, slopes play an important role in our everyday life. You may walk up a
slope to get to the bus stop or ski down the slope of a mountain. The slope formula,
written
, is a useful tool you can use to calculate the vertical and horizontal
change of a variety of slopes.
4.1.2 Intercepts of Linear Equations
We’ve now seen the usefulness of the intercepts of a line. When we know where a line
crosses the x- and y-axes, we can easily produce the graph or the equation for that line.
When we know one of the intercepts and the slope of a line, we can find the beginning or
predict the end of a process.
4.1.3 Graphing Equations in Slope Intercept Form
The slope intercept form of a linear equation is written as
, where m is the
slope and b is the value of y at the y-intercept. Because we only need to know the slope
and the y-intercept to write this formula, it is fairly easy to derive the equation of a line
from a graph and to draw the graph of a line from an equation.
4.1.4 Point Slope Form and Standard Form of Linear Equations
We’ve learned that linear equations can be written in different forms, depending upon
what we either know or want to know about a line. The point slope form,
, is useful in situations involving slope and the location of one or more
points. The standard form, Ax + By = C, is usually easier to use when we need to make
algebraic calculations. When needs or knowledge change, we can convert an equation
from one form into another.
4.2.1 Parallel Lines
Parallel lines are lines that have the same slope. These lines never intersect and always
maintain the same distance apart. To create a parallel line from an equation, we simply
change the y-intercept.
4.2.2 Perpendicular Lines
Perpendicular lines are lines that cross one another at a 90° angle. They have slopes
that are opposite reciprocals of one another. Unlike parallel lines that never touch,
perpendicular lines must intersect.
4.39
Glossary
coordinates
A pair of numbers that identifies a point on the coordinate plane—
the first number is the x-value and the second is the y-value.
intercept
A point where a line meets or crosses a coordinate axis.
linear equation
An equation that describes a straight line.
parallel lines
Lines that have the same slope and different y-intercepts.
perpendicular
lines
Lines that have opposite reciprocal slopes.
point-slope
formula
A form of linear equation, written as
, where m
is the slope and (x1, y1) are the co-ordinates of a point.
rise
Lines that have opposite reciprocal slopes.
run
Horizontal change between two points.
slope formula
The equation for the slope of a line, written as
, where
m is the slope and (x1, y1) and (x2, y2) are the coordinates of two
points on the line.
slope
The ratio of the vertical and horizontal changes between two
points on a surface or a line.
slope-intercept
form
A linear equation, written in the form y = mx + b, where m is the
slope and b is the y-intercept.
slope-intercept
formula
A linear equation, written as y = mx + b, where m is the slope and
b is the y-intercept
standard form
of a linear
equation
A linear equation, written in the form Ax + By = C, where x and y
are variables and A, B, and C are integers.
x-intercept, xintercepts
The point where a line meets or crosses the x-axis.
y-intercept, yintercepts
The point where a line meets or crosses the y-axis.
4.40