Download Plotting Points

§2.1 Graphing Equations
The following is the Rectangular Coordinate System, also called the Cartesian Coordinate
System.
y
II
(-,+)
I
(+,+)
Origin
x
III
(-,-)
IV
(+,-)
A coordinate is a number associated with the x or y axis.
An ordered pair is a pair of coordinates, an x and a y, read in that order. An ordered pair
names a specific point in the system. Each point is unique. An ordered pair is written
(x,y)
The origin is where both the x and the y axis are zero. The ordered pair that describes the
origin is (0,0).
The quadrants are the 4 sections of the system labeled counterclockwise from the upper
right corner. The quadrants are named I, II, III, IV. These are the Roman numerals for
one, two, three, and four. It is not acceptable to say One when referring to quadrant I,
etc.
Plotting Points
When plotting a point on the axis we first locate the x coordinate and then the y
coordinate. Once both have been located, we follow them with our finger or our eyes to
their intersection as if an imaginary line were being drawn from the coordinates.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
1
Example:
a) (2,-2)
Plot the following ordered pairs and note their quadrants
b) (5,3)
c) (-2,-5)
d) (-5,3)
e) (0,-4)
y
f) (1,0)
x
Also useful in our studies will be the knowledge of how to label points in the system.
Labeling Points on a Coordinate System
We label the points in the system, by following, with our fingers or eyes, back to the
coordinates on the x and y axes. This is doing the reverse of what we just did in plotting
points. When labeling a point, we must do so appropriately! To label a point
correctly, we must label it with its ordered pair written in the fashion – (x,y).
Example:
Label the points on the coordinate system below.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
2
Linear Equation in Two Variable is an equation in the following form, whose solutions
are ordered pairs. A straight line graphically represents a linear equation in two
variables.
ax + by = c
a, b, & c are constants
x, y are variables
x & y both can not = 0
Since a linear equation’s solution is an ordered pair, we can check to see if an ordered
pair is a solution to a linear equation in two variables by substitution. Since the first
coordinate of an ordered pair is the x-coordinate and the second the y, we know to
substitute the first for x and the second for y. (If you ever come across an equation that is not
written in x and y, and want to check if an ordered pair is a solution for the equation, assume that the
variables are alphabetical, as in x and y. For instance, 5d + 3b = 10 – b is equivalent to x and d is
equivalent to y, unless otherwise specified.)
Example:
Is (0,5) a solution to: y = 2x  3 ?
Notice that we said above that the solutions to a linear equation in two variables! A
linear equation in two variables has an infinite number of collinear (solutions on the same
line) solutions, since the equation represents a straight line, which stretches to infinity in
either direction. An ordered pair can represent each point on a straight line, and there are
infinite points on any line, so there are infinite solutions. The trick is that there are only
specific ordered pairs that are the solutions!
Graphing a Linear Equation
Step 1: Choose 3 “easy” numbers for either x or y (easy means that your choice either allows
the term to become zero, or a whole number)
Step 2: Solve the equation for the other value that you did not choose in step one. You
will solve for three values.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
3
Example:
Graph
y = 2x
y = 2x – 3
&
y
x
In this example, I have chosen to have you graph two equations with the same slope and
different y-intercepts, so that I can bring up a concept in graphing called translation.
Translation is a way of describing graphs based upon a general form of a family of
graphs, being moved about the coordinate system. Once we learn the basic concept it
makes graphing all types of equations easier, because we can see their basic characteristic
graph being moved by horizontal, vertical, reflection, and stretching translations. In this
section we will discuss, three of these translations – horizontal, vertical and reflection.
The last example is a vertical translation of the line with slope of 2. Just imagine if the
line y = 2x were a piece of wire and we simply picked it up and moved it down the
vertical axis three units, but nothing else about the line changes.
Not all equations produce a line. Those that don't are called non-linear functions or
equations. Martin-Gay/Greene suggest that we learn to graph these non-linear functions
just as we have with a linear equation. I am going to show you some tips about 5 special
non-linear functions (Martin-Gay/Greene discuss 3 at this time and bring the other 2 up during the next
few sections, so I’ll just get them all out of the way now) and how to graph them. We should also
realize that choosing random points and solving the equation/function for those point will
yield ordered paired solutions, but without the knowledge of what a function’s graph
looks like, we may never get a true picture of the function’s graph. Following are a few
of the non-linear functions are some of their characteristics.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
4
Quadratic (2nd Degree)
Form a type of graph called a parabola
Form of equation we'll be dealing with in are:
y = ax2 + d & y = a(x – c)2 + d
Sign of a determines opens up or down
"+" opens up
"" opens down
The vertex (where the graph changes direction) is at:
(0, d) in
y = ax2 + d
(c, d) in
y = a (x – c)2 + d
Symmetric around a vertical line called a line of symmetry
Goes through the vertex, and has the equation x = “x-coordinate of vertex”
Example:
Graph y = x2, y = -x2 and y = -x2 + 2 on this graph, by making a t-table
of points and while thinking about the vertex, the line of symmetry and 4
points. Use the same points for each graph. After we will discuss the
shape, the translations and how those translations effected the points that
we graphed.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
5
Absolute Value Functions
Form a V shaped graph
Form of equation we'll be dealing with are: y = a | x | + d and y = a | x – c | + d
Sign of “a” determines opens up or down
"+" opens up
"" opens down
The vertex (where the graph changes direction) is at:
(0, d) in
y=a|x|+d
(c, d) in
y=a|x–c|+d
Symmetric around a vertical line called a line of symmetry
Goes through the vertex, and has the equation x = “x-coordinate of vertex”
Example:
Graph y = | x | and y = | x – 2 |
on this graph, by making a t-table
of points and while thinking about the vertex, the line of symmetry and 4
points. Use the same points for each graph. After we will discuss the
shape, the translations and how those translations effected the points that
we graphed.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
6
Cubic Functions
Form a lazy "S" shaped graph
Type we'll deal with are:
y = ax3 + d and y = a(x – c)3 + d
Sign of "a" determines curves up and to right or down and to right
"+" curves up and to right
"" curves down and to right
Point of Inflection is where the graph starts to have opposite slope
(0, d) in
y = ax3 + d
(c, d) in
y = a(x – c)3 + d
Choose opposites on either side of the inflection point to graph
Still has symmetry allowing for easy graphing
Example:
Graph y = x3 & y = x3  1
on this graph, by making a t-table
of points and while thinking about the “inflection point”, and 2 positive
x’s and their opposites. Use the same points for each graph. After we will
discuss the shape, the translations and how those translations effected the
points that we graphed.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
7
Square Root Functions
Looks like half a parabola opening up or down from the x-axis
Only half because principle square root is only consideration
Type we'll deal with are:
y = a x + d and y = a x – c
Sign of "a" tells up or down from x-axis
"+" is up
"" is down
Vertex is still (-c,0) similar to a parabola
Example:
Graph y = √ x , y = x y = -√ x – (-3), x  0
on this graph, by making a t-table of points and while thinking about the
vertex, the line of symmetry and 4 points. Use the same points for each
graph. After we will discuss the shape, the translations and how those
translations effected the points that we graphed.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
8
Reciprocal Functions
Look like a wide parabola in 1st & 3rd or in 2nd & 4th quadrants
Never touch or cross an axis
Because x 0, but as it gets close it is very small which makes f(x) get big
Like wise as x gets big, f(x) gets small but never gets to zero!
Type we'll deal with in this chapter are:
y = a/x
Sign of "a" tells which quadrants
"+" is in 1st & 3rd
"" is in 2nd & 4th
Trick in graphing is choosing both small & large values of x that are
both positive and negative
Example:
Graph
y = 2/x
using 8 points. Use 2 small & 2 large
positive & negative values.
y
x
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
9
§2.2 Introduction to Functions
A relation is any set of ordered pairs. A function is a relation for which every value of
the independent variable (the values that can be inputted) has one and only one value for the
dependent variable (the values that are output, dependent upon those input). All the possible
values of the independent variable form the domain and the values given by the
dependent variable form the range. Think of a function as a machine and once a value is
input it becomes something else, thus you can never input the same thing twice and have
it come out differently. This does not mean that you can't input different things and have
them come out the same, however! Your textbook shows some great pictures of
functions on page 97.
There are many ways to show a function's domain and range. One way is to draw a
picture and show the mapping (each element of the domain maps to only one member of the range in
a function) of the domain onto the range. Another way is to list the domain as a set using
roster or set builder notation and the range as a set using roster or set builder notation. If
the domain and range are both finite, they can be listed together as a set of ordered pairs.
We can also graph a function, which shows both the domain and the range.
2
3
5
15
25
5
These are maps.
Left is f(n)
Rt. Is not an f(n)
2
6
7
Is a Relation a Function?
1) For every value in the domain is there only one value in the range?
a) Looking at a map – If any x have lines to more than one y, then not a function
b) Looking at ordered pairs – If no x’s repeat then it’s a function
c) Looking at a graph – Vertical line test (if any vertical line intersects the graph in more
than one place the relation is not a function)
d) Think about the domain & range values – If input of any x will give different
y’s, then not a function (probably a graph is still best!)
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
10
Example:
a)
Which of the following are functions? What are the domain &
range of each?
{(3,5), (2,5), (4,5)}
b)
y = x, {x| x  0, x}
c)
2
2
2



5
7
9
y
d)
5
15
-15
x
-5
e)
Input (Eye Color)
Output (people in
class)
blue
green
brown
Jorge
Sally
Maria
Katerina
Fritha
Manesh
Tung
Note: When a set is finite it is easiest to use roster form to list the domain & range, but when the sets are
infinite or subsets of the , set builder and interval notation are better for listing the elements.
Function notation may have been discussed in algebra, but if it wasn't you didn't miss
much. It is just a way of describing the dependent variable as a function of the
independent. It is written using any letter, usually f or g and in parentheses the
independent variable. This notation replaces the dependent variable.
f (x)
Read as f of x
The notation means evaluate the equation at the value given within the parentheses. It is
exactly like saying y=!!
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
11
Example:
a)
Evaluate
f(2)
f(x) = 2x + 5 at
Example:
a)
For
h(g) = 1/g
h(1/2)
Example:
Let’s look to our book for evaluating a function represented on a
graph. p. 135 #82, 86, & 90 Let’s also see if we can write this
function’s domain and range.
b)
f( -1/2)
find
b)
h(0)
When using function notation in everyday life the letter that represents the function
should relate to the dependent variable's value, just as the independent variable should
relate to its value.
Example:
a)
b)
Y. Butterworth
The perimeter of a rectangle is P = 2l + 2w. If it is
known that the length must be 10 feet, then the perimeter
is a function of width.
Write this function using function notation
Find the perimeter given the width is 2 ft. Write this
using function notation.
Ch. 2 Notes – Martin-Gay/Greene
12
§2.3 Graphing Linear Functions
Recall from §2.1:
Linear Equation in Two Variable is an equation in the following form, whose solutions
are ordered pairs. A straight line can graphically represent a linear equation in two
variables.
ax + by = c
a, b, & c are constants
x, y are variables
x & y both can’t = 0
Also Recall from Beginning Algebra:
Solving an equation for y is called putting it in slope-intercept form. This is a special
form, a function, with the following properties.
y = mx + b
m = slope
b = y-intercept
The other great thing about this form is that it allows us to use function notation and
eliminate the need to write the dependent variable. Hence, y = mx + b becomes
f(x) = mx + b
since y is a function of x.
We reviewed graphing a linear equation in 2 variables in §2.1, based upon 3 randomly
chosen points. In this section graphing is discussed in terms of the intercepts. We will
still need a 3rd random point, but this allows us to more quickly graph using points. I
won’t be spending a lot of time on this as I feel the “plug and chug” method, whether it is
with 3 random points or intercepts and a random point is still an ineffective method
which wastes our valuable time! However, we do need to know what intercepts are and
how they can be used to graph a line.
An intercept is where a graph crosses an axis. There are two types of intercepts for any
graph, an x-intercept and a y-intercept. An x-intercept is where the graph crosses the xaxis and it has an ordered pair of the form (x, 0). A y-intercept is where the graph
crosses the y-axis and it has an ordered pair of the form (0, y) most often written (0, b).
Finding the Y-intercept (X-intercept)
Step 1: Let x = 0 (for x-intercept let y = 0)
Step 2: Solve the equation for y (solve for x to find the x-intercept)
Step 3: Form the ordered pair (0,y) where y is the solution from step two. [the ordered
pair would be (x, 0)]
Note: When finding both intercepts in a single problem, you are expected to clearly mark you work as: The
x-intercept and The y-intercept and then show work for each under these headings finishing with your final
answer as an ordered pair.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
13
Example:
Find the x & y intercepts and give them as ordered pairs
a)
x  2y = 9
b)
y = 2x
Note1: In part a) will the y-intercept be easy to graph?
Note2: In part b)notice that the x & y intercepts are the same. This is because this is a line through the
origin, so it crosses the x & y axis in the same place. If there is no b in slope-intercept form or c = 0 in std.
form then the line goes through the origin.
To Graph a Linear Function
Step 1: Choose 3 values of x (appropriate values)
Now we have 2 special points that are easy to find, the x and y-intercepts & still find the 3rd one
Step 2: Substitute and solve for f(x) [e.g. y] 3 times
Remember that two points make a line, but 3 gives a check! Also recall that the since the
domain and range of a linear function are all real numbers it makes it possible to choose
any value of x and know that there will be a corresponding value for f(x).
Example:
Graph
f(x) = 1/3 x + 1
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
14
Now, we need to discuss another set of special lines. They don't appear to be linear
equations in 2 variables because they are written in 1 variable, but this is because the
other variable can be anything. The lines in question are vertical and horizontal lines.
Here is a table of facts about these special types of lines that you need to know forward,
backward, and upside-down and inside out.
Type
Horizontal
Equation
y = #
Slope
Zero
Type of ordered Pairs
(#1, y), (#2, y), (#3, y);
y agrees with equation &
#1, #2 & #3 can be anything
Vertical
x=#
Undefined (x, #1), (x, #2), (x, #3);
x agrees with equation &
#1, #2,& #3
Intercepts
y-intercept: (0,y)
no x-intercept
No y-intercept
x-intercept: (x, 0)
Vertical Lines
x = a for a  
A vertical line is not a function.
A vertical line has no y-intercept, unless it is the y-axis (x = 0)
x-intercept is value of x
For whatever y chosen, x will equal “a” (thus all y's are valid)
Horizontal Lines
y = b or f(x) = b
for b  
A constant function
A horizontal line has no x-intercept, unless it is the x-axis (y = 0)
y-intercept is the value of y
For whatever x chosen, y will equal “b” (thus all x’s are valid)
Example:
Graph the following two equations on the same coordinate system.
y
a)
x = -1
b)
f(x) = 1/3
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
15
Remember our non-linear functions from the previous sections. In this section we learn
that they also have y-intercepts. The y-intercept of the general function, for all of our
functions except the reciprocal function, is the origin, just as it is for the general form of a
linear equation in 2 variable (that general function is y = ax, btw). When we make a vertical
translation (that’s adding in the d, in all the functions above), then the y-intercept is whatever the
“d” is. This is just like a straight line where we see that when written in function notation
(slope-intercept form) that the constant that is added to the independent variable is the yintercept. The reason that this is the case is because when we let x = 0, this makes the
variable term “go away” and we are simply left with the constant, and we know that
mathematically, finding the y-intercept is letting x = 0 and solving for y, so all the
concepts begin to come together here.
Example:
Match the following 3 functions with their graphs
a)
f(x) = -x2
b)
f(x) = x2 + 2
c)
i)
^
^
ii)
iii)
v
Y. Butterworth
f(x) = -x2 – 3
v
Ch. 2 Notes – Martin-Gay/Greene
v
v
16
§2.4 The Slope of a Line
Slope is the ratio of vertical change to horizontal change.
m = rise = y2  y1 = y
run
x2  x1
x
Rise is the amount of change on the y-axis and run is the amount of change on the
x-axis.
A line with positive slope goes up when viewing from left to right and a line with
negative slope goes down from left to right. When asked to give the slope of a line, you
are being asked for a numeric slope found using the equation from above. The sign of
the slope indicates whether the slope is positive or negative, it is not the slope itself!
Knowing the direction that a line takes if it has positive or negative slope, gives you a
check for your calculations, or for your plotting.
3Ways to Find Slope
1) Formula given above
Example:
Use the formula to find the slope of the line through (5,2) & (-1,7)
2) Geometrically using m = rise/run
Choose points, create rise & run triangle, count & divide
Example:
#70 p.161 Martin-Gay/Greene
3) From the slope-intercept form of a linear function
y = mx + b, where m, the numeric coefficient of x is the slope
Solve the equation for y, give numeric coeff. of x as the slope (including the sign)
Example:
Find the slope of the line
2x  5y = 9
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
17
Special Lines
We have already discussed horizontal and vertical lines, but now we need to discuss them
in terms of their slope.
Horizontal Lines, recall, are lines that run straight across from left to right. A
horizontal line has zero slope. This is so since it has zero vertical change (rise) and zero
divided by anything is zero.
m = rise = 0 = zero
run
#
Example: Find the slope of the horizontal line through the points
(0,5);(10,5)
Note: It is always acceptable to use the formula to find the slope when given points, but if the ycoordinates are the same then the line is horizontal and has zero slope and if the x-coordinates are the
same the line is vertical (recall my table from the last section) and has undefined slope.
Vertical Lines, recall, are lines that run straight up and down. A vertical line has
undefined slope. This is so since it has zero horizontal change (run) and anything divided
by zero is undefined.
m = rise = # = undefined
run
0
Note: Some authors will use the very weak, no slope, to mean either undefined or zero slope. I will not
accept "no slope" for either answer. The reason for this is that no slope can be translated to mean none
or zero and a vertical line does not have zero slope, it has undefined slope, as you will see from the
following example! Also, since this confusion exists, I want it eliminated completely!!
Example: Find the slope of the line below
y
x
3
Note: When finding the slope of a horizontal or vertical line that has been graphed, the formula approach
can still be used, but it is easier to remember that a vertical line has undefined slope and a horizontal line
has zero slope.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
18
Let's go back to the discussion of graphing a line. There are 3 methods, let's recap them
and then focus on the one of interest.
Graphing a Line (A linear equation in two variables)
Method 1: Choose 3 random x's and solve for y and graph the 3 ordered pairs
Method 2: Find x & y intercepts & a third point as in Method 1 and graph
Method 3: Graph y-intercept & use geometric approach of slope to get 2nd & 3rd points.
Slope & y-intercept come from slope-intercept form of line.
Way back in Pre-Algebra you may have learned Method 1 and we reviewed it earlier in
this chapter, and then you learned about the 2 special points – the intercepts and how that
is just method 1 with a couple of points that are pretty easy to find. Now we will focus
our attention on Method 3.
Graphing a Line Using Slope-Intercept Form
Step 1: Put the equation into slope-intercept form (solve for y)
Step 2: Plot the y-intercept point (0, b)
Step 3: Count up/down and over left/right from the intercept point as indicated by the
slope, to find a second point.
Step 4: Repeat Step 3, with a different iteration of the slope (e.g. +/ is same as /+ or +/+ is
same as /). If you went up and left then go down and right, and if you went up and right then go
down and left.
Example:
Graph the following using the method just described
2x + 3y = -9
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
19
Special Pairs (Groups) of Lines
There are two types of special pairs. There are perpendicular lines and parallel lines.
Perpendicular Lines are lines that meet at right (90) angles. Perpendicular lines have
slopes that are negative reciprocals of one another.
Example:
Give the slope of the line perpendicular to each of the following.
a)
y = 1 /2 x + 3
b)
2x + 3y = 9
c)
Thru the points
(0, 3) & (5, 2)
Parallel Lines are lines that never cross. At all points, parallel lines are equidistant.
Parallel lines have the same slope, but they do not have the same y-intercept. These are
lines that are vertical translation of the same line.
Example:
a)
y =
c)
-2
Give the slope of the line parallel to each of the following:
/3x – 3
b)
5x + 4y = 9
Thru the points
(5, -1) & (-2, -3)
Parallel or Perpendicular?
Based upon the slope of the line, we can judge whether or not a pair (group) of lines are
parallel or perpendicular.
Assessing Whether Perpendicular or Parallel
Step 1: Find the slope of the lines, [if equation then put into slope-intercept form (solve for y)]
Step 2: a) For parallel lines -- Are the slopes the same?
b) For perpendicular lines -- Are the slopes negative reciprocals?
Note: The product of negative reciprocals is –1, or take the reciprocal and then the opposite
of one slope and see if it is the other slope.
Example:
Are the lines parallel, perpendicular or neither? Why or why not?
a)
2x  y = -10
b)
x + 4y = 7
c)
f(x) = 5x – 6
2x + 4y = 2
2x = -5y + 3
g(x) = 5x + 2
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
20
Applications of Slope
The slope of a line is the same thing as pitch of a roof and the grade of a climb. It is
exactly the same calculation for pitch as for slope and in grade it is simply converted to a
percentage.
Example:
From the middle of the ceiling to the point of the roof (apex of the roof) it
is 5 feet. From the middle of the ceiling to the outer wall, where the roof
connects, is 10 feet. Find the pitch of the roof.
Example:
The train climbed 2580 vertical meters from the bottom of the hill to the
top, but the climb took 6450 horizontal meters. What is the grade of the
climb?
Example:
The following example came from p. 207, Beginning Algebra, 9th Edition,
Lial, Hornsby and McGinnis
That’s the y-intercept!
Slope
!
Note: Remember our discussions about linear equations in Ch. 1.7, this is an example of one of those
linear equations. The baseline is the y-intercept and the “Amount per” is the slope. This is also known as
a rate of change.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
21
§2.5 Equations of Lines
The equation for a line can be written in 3 different forms. First we learned the standard
form, then we introduced the slope-intercept form and finally we'll learn the point-slope
form. Each way of writing the equation has its drawbacks and its benefits, but the slopeintercept is the most informative and therefore the way that we most typically write the
equation for a line. We will start out learning how to write the equation for a line by
using this form. If we have any of the following scenarios we can use slope-intercept
form.
Slope-Intercept Form
Scenario 1: We have the slope and the intercept both given (intercept may be given as an
ordered pair (0, b))
Scenario 2:
We have two points and one is the intercept point (we can calculate the slope
from the formula)
Scenario 3:
We have a visual line and we can determine two integer ordered pairs, one
of which is the y-intercept (you can’t guess)
Under scenario 1 we have the easiest case. All we have to do is to plug in the slope for
“m” and the intercept for “b” (if it is an ordered pair, pick off the y-coordinate to use as b).
Recall that the general form of the equation in slope-intercept form is:
y = mx + b
m = slope
b = y-intercept
Here are some Scenario 1 examples:
Example:
a)
Use the given information to write an equation for the line described in
slope-intercept form.
m = 2 and b = 3
b)
m = 0 & (0,2/3)
c)
m = undefined
& ( -1/2, 0)
Note: Parts b & c are special cases since they are the slopes for horizontal and vertical lines. In the case
of a horizontal line (slope is zero) there is no problem in plugging into the slope-intercept form, except that
it wastes time. However, you can’t plug into the slope-intercept form for a vertical line since a vertical line
is not a function! You really need to study the facts from the table in §2.1 so that you can recognize a
vertical or horizontal line and give it’s equation no matter what information that you are given!
Under scenario 2 we have a little more work, but it still isn't bad. All we must do is
calculate the slope and then plug into the slope-intercept form as described under the first
scenario.
Example:
Find the slope of the following lines described by the points.
a)
(0, 5) & (-1, 7)
b)
(2, 4) & (0, 0)
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
22
c)
(2, -5) & (2, 0)
d)
(0, 7) and (5, 7)
Note: The last three examples are special cases. B) is a line through the origin, C) is a vertical
line and D) is a horizontal line. C) is the only one that doesn’t fit the scenario, but I threw in the xintercept point to throw you off!
Scenario 3 is just about the same as scenario 2 except we will find the slope by visual
inspection.
Example:
Give the equation of the line shown below.
y
x
Now, let me give you one of each type to try on your own!
Your Turn
Example:
Find the equation of each of the following lines.
a)
m = ½ & b = 4
b)
m = -5 & thru (0, -1)
c)
Y. Butterworth
Thru (-1, 5) & (0, 4)
d)
Ch. 2 Notes – Martin-Gay/Greene
Thru (2,0) and (2, 9)
23
e)
m = 0 & (2, 1)
y
f)
If there is no y-intercept given (or if it is not an integer ordered pair) then we must use the
point-slope form. There are also 3 scenarios here. They are as follows:
Point-Slope Form
Scenario 1: You are given the slope & a point that isn’t the y-intercept – not (0, b)
Scenario 2: You are given two points neither of which is the y-intercept, (0, b)
Scenario 3: You are given a graph & the y-int. isn’t an integer ordered pair.
You need only plug into the point-slope form:
y  y1 = m(x  x1)
m = slope
(x1, y1) is a point on the line
x & y are variables (don’t substitute for those)
Under scenario 1 our job is the easiest!
Example:
Find the equation of the line described by the point and slope given.
a)
(-2, 5) m = -1
b)
(1, -3), m = ½
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
24
Example:
In the following case, why can't point-slope form be used to write the
equation of the line? Why doesn't it matter?
(0, -5), m = undefined
Scenario 2 just increases the number of steps in the process. We must find the slope in
addition to plugging into the point-slope form and solving the equation for y.
Example:
a)
Find the equation of each line through the given points. Use the
point-slope form.
(5, 2) and (2, 5)
b)
(2, 0) and (8, -2)
c)
(-3, -1) and (-4, 2)
c)
(1/2, 5) and (1, 1)
Scenario 3 just makes us visually find the slope and then pick a point from the graph so
that we can use the point-slope form.
Example:
Find the slope of the line shown below.
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
25
Now I’ll give you a chance to try these type of problems too.
Your Turn
Example:
Give the equation of the lines described below using point slope form.
a)
m= 5 thru (5, 2)
b)
Thru (2, -5) & (-2, 7)
c)
y
x
x
Writing An Equation with Special Requirements
Finally, we need to discuss how to write the equation of a line given certain requirements.
Requirement 1 involves the equation of a line that is perpendicular or parallel and a
point that lies on the line for which you are graphing an equation. When you have these
requirements, you can easily find the slope and then use the point-slope form to give the
equation of your new line. Let’s look at a couple of these now.
Example:
Find the equation of the line described.
a)
Parallel to y = 2/3x + 9 through (-9, 7)
b)
Perpendicular to 2x + y = 3 through (4, -3)
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
26
Requirement 2 involves vertical and horizontal lines. Remember the table that I gave
you on page 15? You need to review the equations, slopes and how all ordered pairs look
on horizontal and vertical lines for these examples.
Example:
Find the equation of the line described.
a)
Parallel to the line y = 7 through the point (2, -1/4)
Note: That is parallel to a horizontal line so it is also horizontal and therefore the only thing I’m
interested in is the y-coordinate of my point since the equation looks like y = y-coordinate of the ordered
pair.
b)
Perpendicular to the line y = -2/5 through the point (7, -3)
Note: That is perpendicular to a horizontal line so it is vertical and therefore the only thing I’m interested
in is the x-coordinate of my point since the equation looks like x = x-coordinate of the ordered pair.
c)
With m = 0 through the point (2, -178)
Note: We’ve already seen this type, but just to reiterate – the slope is zero so you know that it is a
horizontal line, and therefore its equation must look like y= y-coordinate of the point.
d)
Through the point (1 5/8, 0) with undefined slope.
Note: We’ve already seen this type, but just to reiterate – the slope is undefined so you know that it is a
vertical line, and therefore its equation must look like x= x-coordinate of the point.
Now, I’ll give you some to try on your own. Most of these should take a few seconds,
because they deal with the special cases. There are only 2 that should require a little
longer to figure out. We may or may not have class time to do these and they may be
assigned as homework.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
27
Your Turn
Example:
Find the slope of the following lines.
a)
Parallel to the line y = 4 and passing through (2,-2)
b)
Perpendicular to x = 1 and passing through (8,111)
c)
Perpendicular to 3x + 6y = 10 through (2,-3)
d)
Vertical through (-1000, 2)
e)
Horizontal through (1239,1/4)
f)
With slope, -4; y-intercept, -2
g)
With undefined slope through (-3, 1)
h)
With zero slope through (1/3,7.8)
i)
Through (5,9) parallel to the x-axis
j)
Through (4.1,-92) perpendicular to the x-axis
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
28
Applications of Slopes and Finding Equations
Remember our application problems where we were given the baseline and the rate of
change? Well, those times are gone in this section! Instead, you’ll be given information
for 2 independent values and their corresponding dependent values. From this
information you will have two ordered pairs, and you will be able to build the linear
equation based on the slope (rate of change) and y-intercept (baseline), using methods that we
have just developed.
Example:
Let’s look at problem #78 on p. 172 of Martin-Gay/Greene’s text
The value of a building bought in 1990 appreciate, or increases as time
passes. Seven years after the building was bought, it was worth $165,000;
12 years after it was bought, it was worth $180,000.
a)
If this relationship between years past 1990 and value
continues in a linear fashion, write an equation describing
it.
b)
Use this equation to estimate the value of the building in
the year 2010.
Note: The baseline is the value in the year 1990. The rate of change is the increase per year over the
original price in 1990.
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
29
§3.6 Graphing Linear Inequalities in Two Variables
I always cover this section with graphing, because it is simply an application of graphing.
A linear inequality in two variables is the same as a linear equation in two variables, but
instead of an equal sign there is an inequality symbol (, , , or ).
Ax + By  C
A, B & C are constants
A & B not both zero
x & y are variables
Determining if an ordered pair is the solution set to a linear inequality is just like
determining if it is a solution set to linear equality; we must evaluate the inequality at the
ordered pair and see if it is a true statement. If it is a true statement, then the ordered pair
is a solution, and if it is false then it is not a solution. The solutions are even more
infinite than those solutions for a linear equation in 2 variables.
Example:
a)
Determine if the following ordered pairs are solutions to
7y + 2  -2x
(-1, 0)
b)
(-1,-1)
c)
(-1,1)
Graphing a Linear Inequality in Two Variables
Step 1: Solve the equation for y (don't forget that the sense of the inequality will reverse if
multiplying or dividing by a negative.)
Step 2:
Step 3:
Graph the line y = ax + b, using at least 2 labeled ordered pairs
a) If  or  then line is solid
b) If  or  the line is dotted
When in slope-intercept form, shade the region as indicated by the inequality
a) Choose a checkpoint in the region that you shade and put it into the original
equation to make sure it creates a true statement (this is really to check that you
correctly flipped your inequality if you had to mult./divide by a negative in put the
equation into slope intercept form)
Step 4:
Label the solution.
Example:
Graph y  2x + 4
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
30
Example:
Graph
2x + y  -1
y
x
Example:
Graph
3x  4y  12
y
x
Y. Butterworth
Ch. 2 Notes – Martin-Gay/Greene
31