Now that we know what equations are, how do we go about visualizing them? For an equation with two variables, $x$ and $y$, we need a graph with two axes: an $x$-axis and a $y$-axis. We will use the Cartesian plane, in which the $x$-axis is a horizontal line and the $y$-axis is a vertical line. Where the two axes cross is called the origin.

Graphing an Equation in Two Variables

Let's start with the following equation:

$y=2x-3$

We'll start by choosing a few $x$-values, plugging them into this equation, and solving for the unknown variable $y$. After creating a few $x$ and $y$ ordered pairs, we will plot them on the Cartesian plane and connect the points.

For the three values for $x$, let's choose a negative number, zero, and a positive number so we include points on both sides of the $y$-axis:

• If $x=-2$, then $y=-7$. We plot the point $(-2,-7)$.
• If $x=0$, then $y=-3$. We plot the point $(0,-3)$.
• If $x=2$, then $y=1$. We plot the point $(2,1)$.

Now we can connect the dots to visualize the graph of the equation:

Graph of $y=2x-3$

The equation is the graph of a line through the three points found above. The line continues on to infinity in each direction, since there is an infinite series of ordered pairs of solutions.

Graphing Non-Linear Equations

Example 1

What graph will the following equation make?

$x^{2}+y^{2} = 100$

Let's figure it out by choosing some points to plot.

First, let's try $x=0$:

$\begin{aligned} (0)^{2}+y^{2} &= 100 \\ y^{2} &= 100 \\ \sqrt{y^{2}}&=\sqrt{100} \\ y &= \pm10 \end{aligned}$

So we plot $(0,10)$ and $(0,-10)$. 

Note that we don't always have to choose values for $x$. For example, let's now try setting $y=0$. 

Through the same arithmetic as above, we get the ordered pairs $(10,0)$ and $(-10,0)$. Plot these as well.

We still don't have enough points to really see what's going on, so let's choose some more. Let's try solving for $y$ when $x=6$:

$\begin{aligned} (6)^2+y^2&=100 \\ 36+y^2&=100 \\ 36+y^2-36&=100-36 \\ y^2&=64 \\ y&=\pm 8 \end{aligned}$

So that's two new points: $(6,8)$ and $(6,-8)$. We get similar results with $x=-6$, to get $(-6,8)$ and $(-6,-8)$.  

Now you can begin seeing that we're drawing a circle with a radius of 10:

Graph of $x^2+y^2=100$

This is a graph of a circle with radius 10 and center at the origin.

Example 2

Let's try another example. This time let's use the following equation:

$ y=-x^{2}+9$

Again, let's plug in some numbers and begin plotting points.

Input values (for the independent variable $x$) from -2 to 2 can be used to obtain output values (the dependent variable $y$) from 5 to 9.  Connect these points with the best curve you can, and you'll discover you've drawn a parabola.

Graph of $y=-x^2+9$

This graph is of a parabola (a U-shaped open curve symmetric about a line). Parabolas can open up or down, right or left; they also have a maximum or minimum value.