Algebra 1 Ch 4.2 – Graphing Linear Equations. Objective Students will graph linear equations using a table. Students will graph linear equations using.

Algebra 1Algebra 1

Ch 4.2 – Graphing Linear Ch 4.2 – Graphing Linear EquationsEquations

ObjectiveObjective

Students will graph linear equations Students will graph linear equations using a table.using a table.

Students will graph horizontal and Students will graph horizontal and vertical linesvertical lines

VocabularyVocabulary A A linear equationlinear equation in two variables is an in two variables is an

equation in which the variables appear in equation in which the variables appear in separate terms and neither variable separate terms and neither variable contains an exponent other than 1.contains an exponent other than 1.

The The solutionsolution to linear equations are to linear equations are ordered pairs which makes the equation ordered pairs which makes the equation true.true.

The The graph of an equationgraph of an equation in x and y is in x and y is the set of all points (x, y) that are solutions the set of all points (x, y) that are solutions of the equation.of the equation.

Example #1Example #1 This is an example of a linear This is an example of a linear

equationequationy = x + 8y = x + 8

All linear equations are functions. All linear equations are functions. That is the value of y (output) is That is the value of y (output) is determined by the value of x (input)determined by the value of x (input)

The linear equation in two variables The linear equation in two variables can also be called the can also be called the rulerule. .

In this case whatever x is plus 8 will In this case whatever x is plus 8 will give you the value of ygive you the value of y

Graphing Graphing Since the results of a linear equations can Since the results of a linear equations can

be expressed as ordered pairs, the linear be expressed as ordered pairs, the linear equation can be graphed.equation can be graphed.

When a linear equation is graphed, When a linear equation is graphed, all all points on the line represent the solution points on the line represent the solution setset of the linear equation. of the linear equation.

There are a number of ways to find the There are a number of ways to find the solution to a linear equation…for today’s solution to a linear equation…for today’s lesson we will look at creating a table of lesson we will look at creating a table of the solutions…the solutions…

TablesTables To create a table of solutions to a To create a table of solutions to a

linear equation do the following:linear equation do the following:1. Choose a minimum of 3 values for x2. Substitute the values of x into the

linear equation3. Simplify to find the value of y4. Write the solutions as ordered pairs

Let’s look at an example…Let’s look at an example…

Example #2Example #2y = 2x – 1 y = 2x – 1

xx y = 2x – 1 y = 2x – 1 yy (x, y)(x, y)

00 y = 2(0) – y = 2(0) – 1 1

- 1 - 1 (0, - 1)(0, - 1)

11 y = 2(1) – y = 2(1) – 11

11 (1, 1)(1, 1)

22 y = 2(2) – y = 2(2) – 11

33 (2, 3)(2, 3)

1. Choose a minimum of 3 values for x

2. Substitute the value of x into the equation

3. Simplify to determine the value of y

4. Write as an ordered pair

CommentsComments How do you choose the value of x?How do you choose the value of x? You can use any number for the value of You can use any number for the value of

x…however, you can make your life easy by x…however, you can make your life easy by choosing -1, 0, and 1, because you can do choosing -1, 0, and 1, because you can do mental math and they are easy to graph.mental math and they are easy to graph.

You choose a minimum of 3 numbers You choose a minimum of 3 numbers because when graphed the expected result because when graphed the expected result is a straight line. If you don’t get the is a straight line. If you don’t get the straight line, then you have to go back and straight line, then you have to go back and check your calculations… check your calculations…

GraphingGraphing Use the ordered pair from the table Use the ordered pair from the table

to graph the linear equation.to graph the linear equation. Again…when graphing the result Again…when graphing the result

should be a straight line…should be a straight line… Any point (ordered pair) on that line Any point (ordered pair) on that line

will be a solution to the linear will be a solution to the linear equation…equation…

(x, y)(x, y)

(0, - 1)(0, - 1)

(1, 1)(1, 1)

(2, 3)(2, 3)

x

y

(0, -1)

(1,1)

(2,3)

y =

2x

– 1

y =

2x

– 1

More Comments…More Comments… Once you get the hang of it…this Once you get the hang of it…this

concept is really easy….concept is really easy…. However, like everything else to become However, like everything else to become

proficient you will need to practice…proficient you will need to practice… It’s ok if you make a mistake…you can It’s ok if you make a mistake…you can

fix it…fix it… The goal here is to keep an open mind The goal here is to keep an open mind

and try…and try… Let’s look as some special equations Let’s look as some special equations

that produce horizontal and vertical that produce horizontal and vertical lines…lines…

Linear EquationsLinear Equations

All linear equations can be written in All linear equations can be written in the form:the form:

Ax + By = CAx + By = C This form is called the This form is called the standard formstandard form

of an equation.of an equation. At this level you are At this level you are requiredrequired to know to know

and be able to manipulate this form and be able to manipulate this form of an equationof an equation

Standard FormStandard Form

Ax + By = CAx + By = C In the standard form of an equation:In the standard form of an equation:

A is the coefficient of xA is the coefficient of x B is the coefficient of yB is the coefficient of y C represents the constantC represents the constant

We talked about coefficients and We talked about coefficients and constants in a previous lessonconstants in a previous lesson

Example #3Example #3 The equation 3x – 4y = 12 is an example The equation 3x – 4y = 12 is an example

of an equation written in standard form.of an equation written in standard form. As we have done in a previous lesson, we As we have done in a previous lesson, we

can write the equation in function form by can write the equation in function form by transforming the equation as follows:transforming the equation as follows:

3x – 4y = 123x – 4y = 12

-3x -3x-3x -3x

– – 4y = -3x + 124y = -3x + 12

– – 44 – – 44

y = - ¾ x + 3y = - ¾ x + 3

Standard Form

Function Form

Horizontal LinesHorizontal Lines

In the standard form of an equation Ax + In the standard form of an equation Ax + By = C, When A=0 the equation reduces to By = C, When A=0 the equation reduces to By = C and the graph will be a By = C and the graph will be a horizontalhorizontal line.line.

We often see this illustrated as the We often see this illustrated as the equation y = b.equation y = b.

In this instance, the equation has In this instance, the equation has no x-no x-valuevalue and the y-value is always the same and the y-value is always the same number so that when the y-value is number so that when the y-value is graphed a graphed a horizontalhorizontal line is produced. line is produced.

Example #4 – Horizontal Example #4 – Horizontal LineLine

Graph the equation y=2Graph the equation y=2 In this instance there is no x-value. All In this instance there is no x-value. All

the y-values = 2the y-values = 2 To plot this line, starting at 0, go up 2 To plot this line, starting at 0, go up 2

spaces on the y-axis and draw a spaces on the y-axis and draw a horizontal line (as shown in the next horizontal line (as shown in the next slide)slide)

Example #4 (Continued)Example #4 (Continued)

x

y

y=2y=2

y=2y=2

CommentsComments

Notice that when you graph the line, Notice that when you graph the line, the line is the line is perpendicularperpendicular to the to the y-axisy-axis..

A A common errorcommon error that students make that students make when graphing an equation like y=2 is when graphing an equation like y=2 is that they draw the line parallel to the y-that they draw the line parallel to the y-axis. axis. That is incorrect!That is incorrect!

A way to avoid this error is to actually A way to avoid this error is to actually plot the point before you draw the line.plot the point before you draw the line.

Vertical LinesVertical Lines

In the standard form of an equation Ax + In the standard form of an equation Ax + By = C, When B=0 the equation reduces By = C, When B=0 the equation reduces to Ax = C and the graph will be a to Ax = C and the graph will be a vertical vertical line.line.

We often see this illustrated as the We often see this illustrated as the equation x = aequation x = a

In this instance, the equation has no In this instance, the equation has no y-y-valuevalue and the x-value is always the same and the x-value is always the same number so that when the x-value is number so that when the x-value is graphed a graphed a verticalvertical line is produced line is produced

Example #5 – Vertical LineExample #5 – Vertical Line

Graph the equation x = -3Graph the equation x = -3 In this instance there is no y-value. In this instance there is no y-value.

All the x-values = -3All the x-values = -3 To plot this line, starting at 0, go 3 To plot this line, starting at 0, go 3

spaces to the left on the x-axis and spaces to the left on the x-axis and draw a vertical line (as shown in the draw a vertical line (as shown in the next slide)next slide)

Example #5(Continued)Example #5(Continued)

x

y

x=-3x=-3

x =

-x =

-33

CommentsComments

Notice that when you graph the line, Notice that when you graph the line, the line is the line is perpendicularperpendicular to the to the x-axisx-axis..

A A common errorcommon error that students make that students make when graphing an equation like x=-3 is when graphing an equation like x=-3 is that they draw the line parallel to the x-that they draw the line parallel to the x-axis. axis. That is incorrect!That is incorrect!

A way to avoid this error is to actually A way to avoid this error is to actually plot the point before you draw the line.plot the point before you draw the line.

CommentsComments On the next couple of slides are some On the next couple of slides are some

practice problems…The answers are on the practice problems…The answers are on the last slide…last slide…

Do the practice and then check your Do the practice and then check your answers…If you do not get the same answers…If you do not get the same answer you must question what you did…go answer you must question what you did…go back and problem solve to find the error…back and problem solve to find the error…

If you cannot find the error bring your work If you cannot find the error bring your work to me and I will help…to me and I will help…

Your TurnYour Turn Find 3 different ordered pairs that Find 3 different ordered pairs that

are the solutions to the equationare the solutions to the equation

1.1. y = 3x – 5y = 3x – 5

2.2. y = -2x – 6 y = -2x – 6

3.3. y = ½ (4 – 2x)y = ½ (4 – 2x)

4.4. y = 4( ½ x – 1)y = 4( ½ x – 1)

Your TurnYour Turn Rewrite the equation in function formRewrite the equation in function form

5.5. 2x + 3y = 62x + 3y = 6

6.6. 5x + 5y = 195x + 5y = 19

Create a table of values & graph the Create a table of values & graph the linear equationlinear equation

7.7. y = -x + 4y = -x + 4

8.8. y= -(3 – x)y= -(3 – x)

9.9. x = 9x = 9

10.10. y = -1y = -1

Your Turn SolutionsYour Turn Solutions1.1. (-1,-8), (0,-5),(1, -(-1,-8), (0,-5),(1, -

2)2)2.2. (-1,-4),(0,-6),(1,-8)(-1,-4),(0,-6),(1,-8)3.3. (-1,3),(0,2),(1,1)(-1,3),(0,2),(1,1)4.4. (-1,-6),(0,-4),(1,-2)(-1,-6),(0,-4),(1,-2)5.5. y = -2/3x + 2y = -2/3x + 26.6. y = -x + 19/5y = -x + 19/5

You should have a table with a You should have a table with a minimum of 3 values. When minimum of 3 values. When plotting the line the following plotting the line the following should be true:should be true:

7.7. Your graph should Your graph should cross the y-axis at +4cross the y-axis at +4

8.8. Your graph should Your graph should cross the y-axis at -3cross the y-axis at -3

9.9. You should have a You should have a vertical line at the vertical line at the point x = 9point x = 9

10.10. You should have a You should have a horizontal line at the horizontal line at the point y= -1point y= -1

SummarySummary A key tool in making learning effective is A key tool in making learning effective is

being able to summarize what you learned in being able to summarize what you learned in a lesson in your own words…a lesson in your own words…

In this lesson we talked about In this lesson we talked about Graphing Graphing Linear EquationsLinear Equations Therefore, in your own Therefore, in your own words summarize this lesson…be sure to words summarize this lesson…be sure to include key concepts that the lesson covered include key concepts that the lesson covered as well as any points that are still not clear as well as any points that are still not clear to you…to you…

I will give you credit for doing this lesson…I will give you credit for doing this lesson…please see the next slide…please see the next slide…

CreditCredit I will add 25 points as an assignment grade for you I will add 25 points as an assignment grade for you

working on this lesson…working on this lesson… To receive the full 25 points you must do the To receive the full 25 points you must do the

following:following: Have your name, date and period as well a lesson Have your name, date and period as well a lesson

number as a heading.number as a heading. Do each of the your turn problems showing Do each of the your turn problems showing allall work work Have a 1 paragraph summary of the lesson in your Have a 1 paragraph summary of the lesson in your

own wordsown words Please be advised – I Please be advised – I will notwill not give any credit for work give any credit for work

submitted:submitted: Without a complete headingWithout a complete heading Without showing work for the your turn problemsWithout showing work for the your turn problems Without a summary in your own words…Without a summary in your own words…