Transcript Alg-1---Ch-4.2-Graphing--Linear-Equations

Algebra 1
Graphing Linear Equations
Objective
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You will graph linear equations using a
table.
You will graph horizontal and vertical lines
Vocabulary
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A linear equation in two variables is an
equation in which the variables appear in
separate terms and neither variable contains an
exponent other than 1.
The solution to linear equations are ordered
pairs which makes the equation true.
The graph of an equation in x and y is the set
of all points (x, y) that are solutions of the
equation.
Example #1
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This is an example of a linear equation
y=x+8
All linear equations are functions. That is
the value of y (output) is determined by
the value of x (input)
The linear equation in two variables can
also be called the rule.
In this case whatever x is plus 8 will give
you the value of y
Graphing
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Since the results of a linear equations can be
expressed as ordered pairs, the linear equation
can be graphed.
When a linear equation is graphed, all points on
the line represent the solution set of the linear
equation.
There are a number of ways to find the solution
to a linear equation…for today’s lesson we will
look at creating a table of the solutions…
Tables
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To create a table of solutions to a linear
equation do the following:
1.
2.
3.
4.
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Choose a minimum of 3 values for x
Substitute the values of x into the linear
equation
Simplify to find the value of y
Write the solutions as ordered pairs
Let’s look at an example…
Example #2
1. Choose a minimum
of 3 values for x
y = 2x – 1
x
0
y = 2x – 1
y = 2(0) – 1
y
-1
(x, y)
(0, - 1)
1
y = 2(1) – 1
1
(1, 1)
2
y = 2(2) – 1
3
(2, 3)
2. Substitute the value
of x into the equation
3. Simplify to
determine the
value of y
4. Write as an
ordered pair
Comments
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How do you choose the value of x?
You can use any number for the value of
x…however, you can make your life easy by
choosing -1, 0, and 1, because you can do
mental math and they are easy to graph.
You choose a minimum of 3 numbers because
when graphed the expected result is a straight
line. If you don’t get the straight line, then you
have to go back and check your calculations…
Graphing
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Use the ordered pair from the table to
graph the linear equation.
Again…when graphing the result should
be a straight line…
Any point (ordered pair) on that line will
be a solution to the linear equation…
y
(2,3)
x
(1,1)
(0, -1)
(x, y)
(0, - 1)
(1, 1)
(2, 3)
More Comments…
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Once you get the hang of it…this concept is
really easy….
However, like everything else to become
proficient you will need to practice…
It’s ok if you make a mistake…you can fix
it…
The goal here is to keep an open mind and
try…
Let’s look as some special equations that
produce horizontal and vertical lines…
Linear Equations
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All linear equations can be written in the
form:
Ax + By = C
This form is called the standard form of an
equation.
At this level you are required to know and
be able to manipulate this form of an
equation
Standard Form
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Ax + By = C
In the standard form of an equation:
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A is the coefficient of x
B is the coefficient of y
C represents the constant
We talked about coefficients and
constants in a previous lesson
Example #3
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The equation 3x – 4y = 12 is an example of an
equation written in standard form.
As we have done in a previous lesson, we can
write the equation in function form by
transforming the equation as follows:
3x – 4y = 12
-3x
Standard Form
-3x
– 4y = -3x + 12
–4
–4
y=-¾x+3
Function Form
Horizontal Lines
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In the standard form of an equation Ax + By
= C, When A=0 the equation reduces to By
= C and the graph will be a horizontal line.
We often see this illustrated as the equation
y = b.
In this instance, the equation has no x-value
and the y-value is always the same number
so that when the y-value is graphed a
horizontal line is produced.
Example #4 – Horizontal Line
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Graph the equation y=2
In this instance there is no x-value. All
the y-values = 2
To plot this line, starting at 0, go up 2
spaces on the y-axis and draw a horizontal
line (as shown in the next slide)
Example #4 (Continued)
y=2
y
y=2
x
Comments
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Notice that when you graph the line, the
line is perpendicular to the y-axis.
A common error that students make when
graphing an equation like y=2 is that they
draw the line parallel to the y-axis. That
is incorrect!
A way to avoid this error is to actually plot
the point before you draw the line.
Vertical Lines
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In the standard form of an equation Ax + By
= C, When B=0 the equation reduces to Ax
= C and the graph will be a vertical line.
We often see this illustrated as the equation
x=a
In this instance, the equation has no y-value
and the x-value is always the same number
so that when the x-value is graphed a
vertical line is produced
Example #5 – Vertical Line
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Graph the equation x = -3
In this instance there is no y-value. All
the x-values = -3
To plot this line, starting at 0, go 3 spaces
to the left on the x-axis and draw a
vertical line (as shown in the next slide)
Example #5(Continued)
x=-3
y
x = -3
x
Comments
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Notice that when you graph the line, the
line is perpendicular to the x-axis.
A common error that students make when
graphing an equation like x=-3 is that
they draw the line parallel to the x-axis.
That is incorrect!
A way to avoid this error is to actually plot
the point before you draw the line.
Comments
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On the next couple of slides are some practice
problems…The answers are on the last slide…
Do the practice and then check your
answers…If you do not get the same answer
you must question what you did…go back and
problem solve to find the error…
If you cannot find the error bring your work to
me and I will help…
Your Turn
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1.
2.
3.
4.
Find 3 different ordered pairs that are
the solutions to the equation
y = 3x – 5
y = -2x – 6
y = ½ (4 – 2x)
y = 4( ½ x – 1)
Your Turn
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5.
6.
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10.
Rewrite the equation in function form
2x + 3y = 6
5x + 5y = 19
Create a table of values & graph the
linear equation
y = -x + 4
y= -(3 – x)
x=9
y = -1
Your Turn Solutions
1.
2.
3.
4.
5.
6.
(-1,-8), (0,-5),(1, -2)
(-1,-4),(0,-6),(1,-8)
(-1,3),(0,2),(1,1)
(-1,-6),(0,-4),(1,-2)
y = -2/3x + 2
y = -x + 19/5
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7.
8.
9.
10.
You should have a table with a
minimum of 3 values. When
plotting the line the following
should be true:
Your graph should cross
the y-axis at +4
Your graph should cross
the y-axis at -3
You should have a vertical
line at the point x = 9
You should have a
horizontal line at the point
y= -1
Summary
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A key tool in making learning effective is being
able to summarize what you learned in a lesson in
your own words…
In this lesson we talked about Graphing Linear
Equations Therefore, in your own words
summarize this lesson…be sure to include key
concepts that the lesson covered as well as any
points that are still not clear to you…
I will give you credit for doing this lesson…please
see the next slide…
Credit
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I will add 25 points as an assignment grade for you working
on this lesson…
To receive the full 25 points you must do the following:
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Have your name, date and period as well a lesson number
as a heading.
Do each of the your turn problems showing all work
Have a 1 paragraph summary of the lesson in your own
words
Please be advised – I will not give any credit for work
submitted:
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Without a complete heading
Without showing work for the your turn problems
Without a summary in your own words…