Graphing Equations

Section 5.2 Graphing Equations

Consider the equation

\begin{equation} x^2+y^2 = 1\tag{5.3} \end{equation}

with two variables \(x\) and \(y\text{.}\) Solutions to the equation are points \((x,y)\) that make the equation TRUE. (See Section 4.1.) The set of all points in the coordinate plane that satisfy the equation is called the graph of the equation. For instance, \(\mathcal{P}(1,0)\) satisfies (5.3)

\begin{equation*} 1^2 + 0^2 = 1 \end{equation*}

so that \(\mathcal{P}(1,0)\) is a point on the graph of the equation. Consider \(\mathcal{Q}\left(0,-1\right)\)

\begin{equation*} 0^2 + \left(-1\right)^2 = 0+1 = 1, \end{equation*}

so that \(\mathcal{Q}\) is on the graph of the equation. Similarly,

\begin{equation*} \left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4}+\frac{3}{4} = 1 \end{equation*}

so that \(\mathcal{R}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\) is also on the graph. If we plot these three solutions in the coordinate plane we obtain just a portion of the graph of \(x^2+y^2=1\text{.}\)

To better understand the graph of this equation, we can rewrite things as

\begin{align*} x^2+y^2 \amp= 1\\ (x-0)^2+(y-0)^2 \amp= 1\\ \sqrt{(x-0)^2+(y-0)^2} \amp= 1 \end{align*}

The left side looks like the Distance Formula (5.1). This says that \((x,y)\) is a solution precisely when the distance between \((x,y)\) and the origin \((0,0)\) is \(1\text{.}\) The set of all points that are one unit from the origin forms a circle centered at the origin with radius \(1\) and it is this circle which is the graph of \(x^2+y^2 = 1\text{.}\) We call this the unit circle.

Figure 5.6. Graph of the unit circle \(x^2+y^2=1\) .

Example 5.7. Horizontal and Vertical Lines.

Describe the graph of the equations \(x=2\) and \(y=0\) in the coordinate plane.

Solution.

The graph of the equation \(x=2\) consists of all points \((x,y)\) with a first coordinate of \(2\text{.}\) These are the points \((2,y)\text{,}\) where \(y\) is any real number. Plotting each of these points results in a vertical line.

The graph of the equation \(y=0\) consists of all points \((x,0)\text{,}\) where \(x\) is any real number. Plotting each of these points results in the horizontal line formed by the \(x\)-axis.

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