Thursday, April 23, 2015

Graphing Lines by Plotting Points

One way to graph lines is by plotting points. You just need to find the points to plot.

Graph:  3x  5y = 15
We begin by solving for the variable y.
 To do this, you need to do some algebra. Be careful here, this is where students often make errors.
After isolating the variable y, we now have an equation where y depends on x. That is, x is the independent variable and y is the dependent variable.

Tip 1: When solving, be sure to divide BOTH terms on the right side by the coefficient of y. In the example above, we divided −5 into the terms −3x and 15 separately.

Now we have an equivalent equation, which will be easier to work with. Get used to fractions, many equations of lines have fractions in them.
Graph:  y = 3/5 x − 3
When graphing by plotting points, teachers typically require that you to plot at least five points. To find these points, you will choose any x-values and then substitute them into the equation to find the corresponding y-values.

Tip 2: Choose some negative x-values, zero, and some positive x-values.

Tip 3: Avoid fractions by choosing x-values wisely.

Since the denominator of 3/5 is 5, we will choose multiples of 5 to avoid fractions. In this case, we choose −5, 0, 5, 10, and 15 for the x-values. Make a table,
t-Chart without y-values
Then substitute these x-values into the equation to find the corresponding y-values.
t-Chart with y-values
The table gives us 5 ordered pairs to plot. Since x-values here are multiples of 5, we choose the tick on the x-axis to represent 5 units. Similarly because the y-values are multiples of 3, we will choose a scale of 3 units on the y-axis.
Graph of y = 3/5 x - 3
Tip 4: Impress your teacher by placing an arrow on either end of the line to indicate that it continues forever.

That’s it!  The general steps are outlined below.

   Step 1: Solve for y so that your equation looks like y = mx + b.
   Step 2: Choose any five x-values. (You only really need two.)
   Step 3: Plug in to find the corresponding y-values.
   Step 4: Plot the points and connect them with a straightedge.

Here is another example.
Graph: −2x − 3y = 6
First solve for y.
Algebra to Solve
Tip 5: Avoid the following common error of dividing only one term.
Common Error
When dividing a binomial by a number you must divide both terms by that number. Next, choose some x-values. Avoid fractions here by choosing −6, −3, 0, 3, and 6. Substituting we have,
And plotting these points we have the graph of  y−2/3 x − 2,
Graph of  y = −2/3 x − 2
Some more examples follow.

Graph: y = 2x − 4 by plotting five points.
   

Graph: y = −x − 2 by plotting five points.

   

Choosing a scale when creating a blank coordinate system will take some thought.  Keep in mind that the scale on the x-axis need not be the same as the scale on the y-axis.

Graph: y = 1/2 x − 6 by plotting five points.

When the coefficient of x is a fraction, choose x-values to be multiples of the denominator so that you might avoid unnecessarily tedious calculations.

Graph:  y = −3/2 x + 6 by plotting five points.

   
But wait there’s more. In the next method [ Plot Using Intercepts ] coming soon, we show an easy two point method for graphing lines.  Read on!