A graph is a picture of all of the solutions to an equation, or a diagram of all of the inputs and outputs to a function.  Both statements have the same general idea, with a slightly different context.

The reason we graph equations is that it provides easy insight into the nature of the solutions and the relationships between inputs and outputs.  The more advanced or applied the mathematics is, the more practical of a tool graphing is.  However, it would be entirely unfair to expect you to make graphs of a complicated system of functions right off the bat.  So, baby steps, so we can learn right!

When you’re asked to graph you are going to sketch a picture of all of the solutions. There are patterns and short-cuts for different types of equations, and you’ll learn each of those in time.  The biggest short-cut (pattern) for linear equations is that the solutions form a line!

Graphing Method #1:  Input/Output Table or t-chart

This method works for any type of graph, which makes it an important tool to have an understand, one that can get you out of trouble if you get stuck.

Because we define x as the input (independent variable), and y as the output (dependent variable), we can just pick some numbers to plug in, perform a calculation, and discover the output.  With linear equations, the math is so easy that sometimes it can be done in your head, entirely!

Let’s see an example.  Let’s fill out a t-chart for y = 3x – 2. 

You have a few choices here for your t-chart.  Some people like the classic t-chart like shown below.  The version on the left is not yet filled out, the one on the right is.

We really only need two points for a linear equation, but more points add a way to check (if they don’t line up, then there’s a mistake).

What we have here is an organized way of finding ordered pairs.  An ordered pair is a solution to an equation or function.  They’re called ordered because they’re alphabetized, and pairs, because there are two values, x and y. 

Each of these points can be plotted on a coordinate plane, as shown below. 

We know this is a linear equation, so we can connect the dots.  That means something though, connecting the dots.  When a line or curve is drawn on a coordinate plane, it means that every single coordinate on that line (or curve) is a solution.  It also means that everything NOT on the line (or curve) is not a solution!

For example, the point (1.8, 3.4) is a solution to our equation.  Let’s check it.

y = 3x – 2 -> (1.8, 3.4)

(3.4) = 3(1.8) – 2

3.4 = 5.4 – 2

3.4 = 3.4

But, any point NOT on the line is not a solution.  For example, (2, 7)

y = 3x – 2 à (2, 7)

(7) = 3(2) – 2

7 = 6 – 2
FALSE

There’s another way to set up a t-chart that is particularly handy when the numbers are trickier, though it looks more like an H-chart.  The benefit is that there’s room in the middle section to do your scratch work.  That way, if your points don’t line up you can find your arithmetic error easily.

Advice:  Always use x = 0 in your t-chart.  Use a mix of positive and negative numbers.  If the coefficient of x (the slope) is a fraction, use values for x that will reduce easily.