# Graphing Equations

Well enough statistics, let’s return to some algebra topics. I’ve done a couple of posts on graphing and I would like to return to that.

So if you remember, a graph which we call the  cartesian coordinate system, is a way of plotting points in the form of (x, y) where x is the horizontal axis coordinate and y is the vertical coordinate. Below is a plot of several points on a graph:

But this is rather boring. Much more interesting is the graph of an equation which is a picture of all the x and y values that satisfy the equation. So if I have an equation y = x + 3, the plot of that equation is below with a few points labelled that show that they do indeed solve the equation, that is, make it true:

For example, the point (1, 4) solves this equation because when I substitute in x = 1 and y = 4, I get a true equation:

${y}\hspace{0.33em}{=}\hspace{0.33em}{x}\hspace{0.33em}{+}\hspace{0.33em}{3}\hspace{0.33em}\Longrightarrow\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}{+}\hspace{0.33em}{3}\hspace{0.33em}\Longrightarrow\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{4}$

Plots of equations can be many shapes, but the concept is always the same: the plot is a picture of all the points that satisfy the equation. Let’s look at

${y}\hspace{0.33em}{=}\hspace{0.33em}{x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{4}$

If you just choose some x values and substitute them in and find the corresponding y values, you can get enough points plotted to show the approximate shape of the graph. For example, if x = 0, then y = -4. So (0, -4) is a point on the graph of this equation. If if x = 2, then y = 0, so (2, 0) is a point on the graph of this equation. Below is the graph of this equation with some points labelled:

Of course, there are an infinite number of points that satisfy this equation like (1.5, -1.75). The point is, this is a picture of all the points that satisfy the equation (within the plot borders of course as the graph goes up forever).

Graphs of equations are very useful in many areas of math, science, and engineering. In my next post, I’ll use a graph to show how it helps visualise a physical process like throwing a ball up in the air.