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Review of Linear Functions
Dr. Philippe B. Laval
Kennesaw State University
January 15, 2003
Abstract
This handout is a quick review of the material students should know
for this class. This material, especially the sections on the slope of a line
and deriving the equation of a line, will be heavily used in Calculus
1
Introduction - X and Y-intercepts
We begin with the simplest of all functions, linear functions. Their graph is a
straight line. Before we give a formal definition of a linear function, we start by
defining two concepts which apply to linear function as well as other functions.
Definition 1 (x-intercept) The x-intercepts of the graph of a function y =
f (x) are the points at which the graph of the function crosses the x-axis.
Definition 2 (y-intercept) The y-intercept of the graph of a function y =
f (x) is the point at which the graph of the function crosses the y-axis.
Since an x-intercept is on the x-axis, its y-coordinate is always 0. So, an
x-intercept is a point of the form (x, 0). To find the x-intercepts, we only need
to worry about finding the x-coordinate. To do so, we set y = 0 in the equation
defining the function, and we solve for x. For example, to find the x-intercepts
of y = 3x − 6, we set y = 0 and solve
0 = 3x − 6
3x = 6
x = 2
so, there is one x-intercept, it is (2, 0). Be aware that a graph may have more
than one x-intercept.
Since a y-intercept is on the y-axis, its x-coordinate is always 0. So, a yintercept is a point of the form (0, y). To find the y-intercept, we only need to
worry about finding the y-coordinate. To do so, we set x = 0 in the equation
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defining the function, and we solve for y. For example, to find the y-intercept
of y = 3x − 6, we set x = 0 and solve
y
y
= 3(0) − 6
= −6
so, the y-intercept is (0, −6). Note that the graph of a function can only have
one y-intercept. why?
2
Definitions
Definition 3 (linear function) A linear function is a function of the form
y = mx + b. This is called the slope-intercept form of a linear function. In this
form, m and b are real numbers, m corresponds to the slope of the line, b is the
y-intercept.
Example 4 Determine the slope and the y-intercept of y − 2 = 3x − 4
First, we need to put the linear function in slope intercept form. This is done
by solving for y.
y − 2 = 3x − 4
y = 3x − 4 + 2
y = 3x − 2
So, the slope is 3, the y-intercept is (0, −2).
Example 5 Determine the slope and the y-intercept of y = x + 3
The equation is already in slope-intercept form. The y-intercept is (0, 3). x is
the same thing as 1x. So, the slope is 1.
Example 6 Determine the slope and the y-intercept of y = 4x
4x is the same as 4x + 0. So, the slope is 4, the y-intercept is (0, 0).
Example 7 Determine the slope and the y-intercept of y = 5
y = 5 is the same as y = 0x + 5. So, the slope is 0, the y-intercept is (0, 5).
Remark 8 (Technology Note) If you wish to experiment with how m and b
affect the way a line looks like, I have designed an applet for this. Either visit,
or click on the link:
http://science.kennesaw.edu/~plaval/applets/Linear1.html??
3
Slope of a Line
Definition 9 (slope of a line)
change in y
change in x
1. The slope of a non-vertical line is
2
rise
=
run
2. If the line is horizontal, its slope is 0 (because the rise is 0).
3. If the line is vertical, its slope is undefined (because its run is 0).
Figure 1 shows the geometrical representation of the slope:With the notation
Figure 1: Slope of a Line
of the picture,
rise
y2 − y1
. We have the following result:
=
run
x2 − x1
Proposition 10 The slope of the line through two points (x1 , y1 )and (x2 , y2 )is
y2 − y1
.
given by the formula
x2 − x1
Example 11 The slope of the line through (2, 1) and (4, 7) is
7−1
=3
4−2
The sign of the slope tells us a lot about the slope. We will remember the
following:
Proposition 12
• A line which rises from left to right has a positive slope (m > 0)
• A line which falls from left to right has a negative slope (m < 0)
• A horizontal line has a slope m = 0
• The slope of a vertical line is undefined.
Remark 13 (Technology Note) If you wish to experiment with the slope of
a line, I have developed two applets which can help you do this.
1. To experiment with how the value of m affects the graph of a linear functin,
click on or visit the link:
http://science.kennesaw.edu/~plaval/applets/Linear1.html
2. To experiment with the notion of slope, click on or visit the link:
http://science.kennesaw.edu/~plaval/applets/LinSlope.html
3
4
Parallel and perpendicular lines
Consider two lines L1 and L2 . Assume the equation of L1 is y = m1 x + b1 and
the equation of L2 is y = m2 x + b2 .
• L1 is said to be parallel to L2 , denoted L1 L2 if the two lines have the
same slope, that is if m1 = m2 .
• L1 is said to be perpendicular to L2 , denoted L1 ⊥ L2 if the product of
the slopes of the two lines is −1, that is if m1 m2 = −1.
Example 14 The line y = 2x + 5 is parallel to y = 2x − 10 because both lines
have the same slope, 2.
Example 15 The lines y = 2x + 5 and x + 2y − 10 = 0 are perpendicular
because the slope of the first line is 2, the slope of the second is − 12 .
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Graphing
5.1
Given two Points on the Line
A line is uniquely determined by two points. To graph a line given two points,
simply plot the points and draw the line through them.
5.2
Given a Point and the Slope
A line is also uniquely determined given a point on the line and its slope. To
graph a line given a point and its slope, proceed as follows:
• Plot the point.
• Using the slope, figure out what the rise and the run are, apply them to
the existing point to find a new point.
• Once the second point is found, draw the line through the two points.
3
Example 16 Graph the line through (1,2) with slope m = . Remember,
4
rise
by definition, slope =
so, here, the rise is 3, the run is 4. This means
run
that when we move 3 units up (the rise) we also move 4 units right (the run).
If we start at (1, 2) and move 3 units up and 4 units right, we end up at
(1 + 4, 2 + 3) = (5, 5) . This is our second point.
Remark 17 In general, when given a point (x1 , y1 ) on a line and the slope of
rise
the line
, to graph the line, one finds a second point on the line by doing
run
(x1 + run, y1 + rise). Once two points are known, the line is easy to graph.
4
5.3
Given the Equation of the Line
There are several possibilities here, they will depend on how the equation is
given to us. The idea is to use the equation to either find two points on the line,
or one point and the slope. Then we can refer to the previous two sections.
5.3.1
Finding one point and the slope.
You will use this option if the equation is already in slope intercept form, or if
you feel it is easy to put the equation in slope-intercept form. In this case, one
has the slope and the y-intercept of the line without having to do any work.
Example 18 Graph the line y = 5x+6. Here, the slope is 5 and the y-intercept
is (0, 6) . Now that we have a point and the slope, we can use the three-second
technique.
5.3.2
Finding two points on the line.
When the equation of the line is known, finding points on the line is very easy.
Simply pick a value for x, plug it in the equation to find the corresponding y
(or vice-versa i.e. pick y and find the corresponding x). Do this twice, you will
have two points.
Example 19 Graph the line 2y + 4x = 8. We pick two values of x at random,
then find the corresponding y. Pick x = 0, then we have
2y + 4 (0) = 8
2y = 8
y = 4
So, (0, 4) is on the line. Next, pick x = 1, then we have
2y + 4 (1)
2y + 4
2y
y
=
=
=
=
8
8
4
2
So, (1, 2) is on the line. Now that we have two points on the line, we can graph
it.
6
Intersection of Lines
To find the intersection of two lines, do the following
1. Put the equation of each line in slope-intercept form.
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2. If the two lines have the same slope, they are parallel and hence have no
intersection. Otherwise, go to the next step.
3. Set the right sides equal and solve for x, this will give the x-coordinate
of the point on intersection.
4. Plug the x value you found in the previous step in one of the equations,
compute the corresponding y-value. This will give you the y-coordinate
of the point of intersection.
Example 20 Find the point of intersection of y = 2x − 5 and 4x + y = 7.
The slope-intercept form of the first equation is y = 2x − 5, for the second
equation it is y = −4x + 7.
Next, we set the right sides equal. We have to solve:
2x − 5 = −4x + 7
6x = 12
x = 2
Next, plug x = 2 in one of the equations, say the first one. We get
y
y
= 2(2) − 5
= −1
So, the point of intersection is: (2, −1)
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Deriving the Equation of a Line
The equation of a line can be written in several forms:
1. Standard form: Ax + By + C = 0
2. Slope - intercept form: y = mx + b. This gives the equation of the line
with y-intercept (0, b) and slope m.
3. Point - slope form: y − y0 = m (x − x0 ), this gives the equation of the
line through (x0 , y0 ) with slope m.
4. Horizontal line: y = b. Gives the equation of the horizontal line through
a point with y-coordinate equal to b.
5. Vertical line: x = a. Gives the equation of the vertical line through a
point with x-coordinate equal to a.
How one derives the equation of a line depends on the information given.
Since a line is uniquely determined given its slope and a point, or two points,
we consider these two cases below.
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7.1
Given a Point and the Slope
Proposition 21 The equation of the line through (x1 , y1 ) with slope m is
y − y1 = m (x − x1 )
This is called the point-slope form of the equation of a line.
Remark 22 The above formula does not work in the case of a vertical line,
since a vertical line has no slope. On a vertical line, all the points have the same
x-coordinate, call it C. The equation of such a vertical line is always of the form
x = C. Once we know a point on a vertical line, we also know the equation of
the line.
Example 23 Find the equation of the line through (1, 2) with slope m = 3.
Here, the given point is (x1 , y1 ) = (1, 2), and m = 3. Substituting in the pointslope formula, we get
y − 2 = 3 (x − 1)
y − 2 = 3x − 3
y = 3x − 1
Example 24 Find the equation of the vertical line through (2, 6).
The equation of a vertical line is always of the form x = C, where C is a
constant. Since (2, 6) is on the line, x = 2.
7.2
Given two Points on the Line
Proceed as follows:
• Use the slope formula to find the slope of the line.
• Use the slope found above, and any one of the given points plus the
point-slope formula to derive the equation.
Example 25 Find the equation of the line through (1, 5) and (2, 11)
• First, we find the slope of the line. Slope=
• Next, we use (1, 5) to find the equation.
y − 5 = 6 (x − 1)
y − 5 = 6x − 6
y = 6x − 1
7
11 − 5
=6
2−1
8
Chapter Review
8.1
Technology Note
Two java applets were introduced in this handout. The student should experiment with them, and know how to use them.
1. To experiment with how the value of m affects the graph of a linear functin,
click on or visit the link:
http://science.kennesaw.edu/~plaval/applets/Linear1.html
2. To experiment with the notion of slope, click on or visit the link:
http://science.kennesaw.edu/~plaval/applets/LinSlope.html
8.2
What the Student Should Know
The student should know the following about linear functions:
• Definition, standard form.
• What is the y-intercept, how do we find it?
• What are the x-intercepts, how do we find them?
• Slope of a linear function, how do we find it? meaning of the sign of the
slope.
• Know how to graph a line given two points, a point and the slope, or the
equation of the line.
• How do we find the equation of a line, given a point and the slope, or two
points?
• Be able to tell if a point is on a line or not.
• Given a word problem involving linear functions, be able to make points
from the data and find the appropriate linear function describing the problem.
8.3
X and Y-intercepts, Slope.
For each case below, tell whether the given function is a linear function. If it is,
put it in standard form, slope - intercept form and point - slope form. Find the
x and y-intercepts and the slope. If it is not a linear function, explain why.
Exercise 26 2x + 3y = 12
Exercise 27 (x + 1) − 2(y + 2) = 4x − 1
Exercise 28 (x + 1)(y + 2) = 1
Exercise 29
y+x=x−1
Exercise 30 y = 500x − 10000
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8.4
Graphing
Graph each function below using the slope and a point. In each case, say which
point you are using, and what the slope is. Then, verify your answer by graphing
the functions on the calculator.
2
Exercise 31 y = x − 5
3
Exercise 32 2y − 5x = 3
Exercise 33 y = x
8.5
Graphing Using the Calculator
Exercise 34 Using your calculator, graph the lines y − 2x + 1 = 0 and y =
−2x + 5. Then, using the ZOOM and TRACE keys, find the coordinates of the
x and y-intercepts as well as the coordinates of the point at which they intersect.
8.6
Finding the Equation of a Line
Do each problem below, explaining each step.
Exercise 35 Find the equation of the line through (2,5) (4, 11)
Exercise 36 Find the equation of the vertical line through (3,6).
Exercise 37 Find the equation of the horizontal line through (7,2)
Exercise 38 Find the equation of the line through (1, 2) parallel to the line with
equation y − 2x − 4 = 0
Exercise 39 Find the equation of the line through (1, 2) perpendicular to the
line with equation 2y + x = 10
8.7
Miscellaneous
Exercise 40 Let F denote the temperature in ◦ F and C denote the temperature
in ◦ C. Knowing that the relation between F and C is linear, and that water
freezes at 0◦ C and boils at 100◦ C, express F as a function of C.
Exercise 41 Let P denote the US population, t denote the number of years.
Assuming the relation between P and t is linear and given that the US population
was 180 millions in 1960 and 250 millions in 1990 express P as a function of
t.
Exercise 42 Find the point of intersection between the lines y + 1 = 3x and
y−x−1 =0
Exercise 43 Find the point of intersection between the lines y − 2x = 5 and
2y − 20 = 4x
Exercise 44 Do # 17, 19, 21, 23, 25, 27 on page 79 of your book
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