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!

The Rectangular Coordinate System

The rectangular coordinate system consists of two real number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. These two number lines define a flat surface called a plane, and each point on this plane is associated with an ordered pair of real numbers (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. The intersection of the two axes is known as the origin, which corresponds to the point (0, 0).
Rectangular Coordinate System

The x- and y-axes break the plane into four regions called quadrants, named using Roman numerals I, II, III, and IV. The ordered pair (x, y) represents the position of points relative to the origin. A few ordered pairs are plotted below:
Quadrants in the Cartesian Coordinate System

This system is often called the Cartesian coordinate system, named after the French mathematician René Descartes (1596–1650). Read about him on Wikipedia, he was a very interesting person!


A linear equation has standard form,
ax + by = c
 
where a, b, and c are real numbers and a and b are not both zero. With numbers a linear equation looks like,
6x  3y = 12
Solutions to linear equations are ordered pairs (x, y) where the coordinates, when substituted into the equation, produce a true statement. For example, to show that (3,2) and (−2, −8) are solutions substitute and then simplify:
Here we can see (3, 2) and (−2, −8) are solutions to the equation. From geometry we know that two points determine a line.  So at this point we can plot these points and then draw a line through them.  The line represents all ordered pair solutions and is called its graph.
Graph of A Line

We have graphed our first line!

Tip 1: The scale on the x-axis does not need to match the scale on the y-axis.  Choose a scale that is convenient. In this case, each tick on the y-axis represented 2 units. However, when you do this you need to be careful when plotting the points.

Tip 2: You do not really need graph paper, but it sure does help. Go to Google and search “free printable graph paper.” Or visit OpenAlgebra.com and click on the link to printable graph paper on the right and print some out.

Tip 3: Use a straightedge – a ruler or ATM card. All lines are straight, do not try to freehand them especially when accuracy is important.

Notice that (−3, 2) is not on the line and so these coordinates should not solve the equation.
Now, the question is, “how did we get those points?”  Read [ Graphing Lines by Plotting Points ] and you will see.
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