LINEAR FUNCTIONS

Linear Equations

Example #1 This is an example of a linear

equationy = 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 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 To create a table of solutions to a

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…

Example #2y = 2x – 1

x y = 2x – 1 y (x, y)0 y = 2(0) –

1 - 1 (0, - 1)

1 y = 2(1) – 1

1 (1, 1)

2 y = 2(2) – 1

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

Comments 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 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…

(x, y)(0, - 1)(1, 1)(2, 3)

x

y

(0, -1)

(1,1)

(2,3)

y =

2x

– 1

More Comments… 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

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

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

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 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

– 4y = -3x + 12

– 4 – 4

y = ¾ x - 3

Standard Form

Function Form

Horizontal Lines

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

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)

x

y

y=2

y=2

Comments

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

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

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

y

x=-3

x =

-3

Comments

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 On the next couple of slides are some

practice problems..

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…

Read 3x:Bring 1 whole sheet of

paper.COPY and ANSWER !

Your Turn Find 3 different ordered pairs that

are the solutions to the equation: (3 ordered pairs for each number). {1,0,-1}

1. y = 3x – 52. y = -2x – 6 3. y = ½ (4 – 2x)4. y = 4( ½ x – 1)

Your Turn Rewrite the equation in function

form5. 2x + 3y = 66. 5x + 5y = 19 Create a table of values & graph the

linear equation7. y = -x + 48. y= -(3 – x)9. x = 910. y = -1

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

2)2. (-1,-4),(0,-6),(1,-8)3. (-1,3),(0,2),(1,1)4. (-1,-6),(0,-4),(1,-2)5. y = -2/3x + 26. y = -x + 19/5

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

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

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

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

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

Summary 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: ½ crosswise I will add 10 points as an assignment grade for you

working on this lesson… To receive the full 10 points you must do the

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

title as a heading. Have a 1 paragraph summary of the lesson in your

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

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

On a ½ crosswise:

Name :Date:

Lesson title

Summary of the lesson