Professor Dave here, let’s graph some lines.
We’ve done quite a bit of algebra up to this point, by learning how to manipulate
and solve different kinds of algebraic equations using a variety of techniques.
But algebra isn’t just about manipulating equations, we will be doing a lot of graphing
as well, because graphs are visual representations of equations that help us comprehend the relationship
between two variables.
Let’s take an incredibly simple equation, Y equals X.
This says that whatever X is equal to, Y is equal to that, too.
So if we make a chart of X values and their corresponding Y values, we get zero, and zero.
One, and one.
Two, and two.
This could continue to infinity in either direction.
Is there some way we can draw a picture that demonstrates this relationship?
Why don’t we draw a grid.
On this grid, which we call the coordinate plane, we can create two axes, the X axis
and the Y axis, which split the plane up into four quadrants.
Those are quadrants one, two, three, and four, represented by roman numerals.
By convention, the X axis is horizontal, and the Y axis is vertical.
Upon these axes, we can draw a physical representation of any algebraic relationship.
For example, if each tick mark is worth one, then we can place all of these pairs of X
and Y values where they belong on the coordinate plane.
If X equals zero and Y equals zero, that would be this point here where the axes cross.
This is called the origin.
So let’s put a dot there.
For one and one, we need to get to an X value of one, which is one to the right, and a Y
value of one, which is up one.
So let’s put a dot here.
We can do the same for two and two, three and three, and so forth.
Each of these points can be described with coordinates called ordered pairs.
These can be represented as a pair of numbers within parentheses, separated by a comma.
That means two comma two will correspond with this point, and we always list the X value
first, followed by the Y value.
If we want to find seven comma four, we start at the origin and go seven to the right, then
up four.
If we want to find negative three comma negative six, we start at the origin and go three to
the left, and six down, as left and down are the negative directions for X and Y respectively.
Now with all these dots pertaining to our original data set, we may as well connect
them, because it’s not just integers that satisfy this relationship.
If X is one point five nine, so is Y.
If X is two hundred sixty seven one hundred twenty fifths, so is Y.
So this line contains all the possible real numbers that satisfy this equation.
In this way, the line represents the equation Y equals X.
Let’s try another, Y equals two X plus three.
Again, we can make a table where we choose any numbers we like for the independent variable,
X, and see what values we get for the dependent variable, Y, which depends on the value of
X for its own value.
When X is zero, Y is three.
If X is one, Y is five.
If X is two, Y is seven.
So we can see the relationship between X and Y begin to emerge, even just by looking at
the numbers, every time X grows by one, Y grows by two, because of this two in the equation.
But this relationship is understood more effectively if we draw the graph.
Here are the first three points we mentioned, with their ordered pairs listed.
Remember it’s the X value first, then the Y value.
We connect the dots, and there is our line.
Now the interesting thing about graphing these equations is that we can use the graph to
make inferences about the system instead of having to do any math.
Say we take this same equation, and we want to know the value of X when Y equals zero.
We can trace along this line until it crosses the X-axis, and right where it touches the
X-axis is where Y must equal zero.
At that point, the X value is right in between negative one and negative two, so it must
be negative three halves.
Of course we can very easily do the algebra to verify this.
Plug in zero for Y, subtract three, divide by two, and we quickly get the solution that
way, but as equations get more complex, it won’t be possible to do such quick math,
and therefore information present on a graph that we can see visually will become more
and more valuable in trying to describe algebraic systems.
Also, graphing equations gives us a new and beautiful perspective on math itself.
We can view Y equals two X plus three as simply an equation, or we can view it as the representation
of a line, a genuine geometric object, determined by all the points in the XY plane that satisfy
this relationship.
The fact that there are many different perspectives in math is not only incredible, but also useful,
as a variety of approaches will better allow us to solve a variety of problems.
We will be graphing quite a bit throughout our study of math, which means we’d better
understand the coordinate plane, so let’s check comprehension.