Chapter 3.2

Graphing Linear equations: 

I. Determining solutions to linear equations

Example 1.  Given the equation  determine if the point (2, 8) is a solution.   

Solution:

Note that when we say (2, 8), this is ordered pair notation for x=2, y=8.  So to determine if (2, 8) is indeed a solution to the given equation, we merely substitute 2 in for x and an 8 in for y to see if it satisfies it.

So, yes, (2, 8) is a solution to the equation .

Example 2.  Given the equation , determine if (2, 1)  is a solution.

Solution:

Substituting 2 in for x and a 1 in for y we get the following:

Since 8 does not equal 9, (2,1) is not a solution.

Example 3.  An equation and two ordered pairs are given.  Show that each pair is a solution of the equation.  Then graph the two pairs to determine another solution.

Solution:

We first begin by verifying that the given points are indeed solutions to the equation.

       

So we have verified that they are solutions.  Now we need to plot the two points and draw a line through them.

 

So another point on the graph appears to be the point (0,2)  and we may (should) verify this.

Example 4.  Graph the equation

Solution: Begin by choosing any two values you want for x—it does not matter what they are.  An easy one is x=0,  and let’s choose x=1 for the second one.  We should choose a third point as kind of a “check point.  Let’s use x=2 Now just plug  in the x=0 into the equation to get the first part of the chart and do likewise with x=2 and x=3.  You should ultimately obtain the chart and graph below: Now we just plot the points  (0, 1), (1, 3), (2, 5).

Now draw a line through these points. (You do not need to show two separate graphs…just do it on one.)

 Got it?

Quick Check:

Click on the equation to see the graph

Graph the following by hand and with the Graphing calculator:

(hint:  Solve for y first)

One of the more difficult things to do when graphing equations with the TI-83 is setting the viewing window. This next example will demonstrate how this is done. 

In future sections you will be expected to determine the viewing window yourself.

Example 5.  Graph the equation using both viewing windows indicated.  Determine which window best shows the shape of the graph.  Pick the one where you can see both the x and y –intercepts.

 with Xscl=1 and Yscl=1

 with Xscl=5 and Yscl=5

Here the notation in the brackets corresponds to the [window] as follows

The Xscl and Yscl just puts the “tic marks” on the axis…they really don’t affect how the graph looks. Lets graph it with the first window:

	Step 1.  Plug in Y1
	Step 2.  Hit the [window] button and enter in the appropriate numbers
	Step 3.  Hit the [graph] key.  You should get the graph on the left.
	That does not quite get the x or y intercepts.  Lets try the other window.
	Step 4.  Input the values for the window of part b.
	Step 5.  Hit the [graph] key.  You should get the graph on the left.
	You will note that both the x and y –intercepts are shown.  So part b gives us the better viewing window.

 Help with Number 67 On page 153