# 2.1 The rectangular coordinate systems and graphs  (Page 3/21)

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Construct a table and graph the equation by plotting points: $\text{\hspace{0.17em}}y=\frac{1}{2}x+2.$

 $x$ $y=\frac{1}{2}x+2$ $\left(x,y\right)$ $-2$ $y=\frac{1}{2}\left(-2\right)+2=1$ $\left(-2,1\right)$ $-1$ $y=\frac{1}{2}\left(-1\right)+2=\frac{3}{2}$ $\left(-1,\frac{3}{2}\right)$ $0$ $y=\frac{1}{2}\left(0\right)+2=2$ $\left(0,2\right)$ $1$ $y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}$ $\left(1,\frac{5}{2}\right)$ $2$ $y=\frac{1}{2}\left(2\right)+2=3$ $\left(2,3\right)$

## Graphing equations with a graphing utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style $\text{\hspace{0.17em}}y=_____.\text{\hspace{0.17em}}$ The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.

For example, the equation $\text{\hspace{0.17em}}y=2x-20\text{\hspace{0.17em}}$ has been entered in the TI-84 Plus shown in [link] a. In [link] b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows $\text{\hspace{0.17em}}-10\le x\le 10,$ and $\text{\hspace{0.17em}}-10\le y\le 10.\text{\hspace{0.17em}}$ See [link] c .

By changing the window to show more of the positive x- axis and more of the negative y- axis, we have a much better view of the graph and the x- and y- intercepts. See [link] a and [link] b.

## Using a graphing utility to graph an equation

Use a graphing utility to graph the equation: $\text{\hspace{0.17em}}y=-\frac{2}{3}x-\frac{4}{3}.$

Enter the equation in the y= function of the calculator. Set the window settings so that both the x- and y- intercepts are showing in the window. See [link] .

## Finding x- Intercepts and y- Intercepts

The intercepts    of a graph are points at which the graph crosses the axes. The x- intercept    is the point at which the graph crosses the x- axis. At this point, the y- coordinate is zero. The y- intercept is the point at which the graph crosses the y- axis. At this point, the x- coordinate is zero.

To determine the x- intercept, we set y equal to zero and solve for x . Similarly, to determine the y- intercept, we set x equal to zero and solve for y . For example, lets find the intercepts of the equation $\text{\hspace{0.17em}}y=3x-1.$

To find the x- intercept, set $\text{\hspace{0.17em}}y=0.$

$\begin{array}{ll}\text{\hspace{0.17em}}y=3x-1\hfill & \hfill \\ \text{\hspace{0.17em}}0=3x-1\hfill & \hfill \\ \text{\hspace{0.17em}}1=3x\hfill & \hfill \\ \frac{1}{3}=x\hfill & \hfill \\ \left(\frac{1}{3},0\right)\hfill & x\text{−intercept}\hfill \end{array}$

To find the y- intercept, set $\text{\hspace{0.17em}}x=0.$

$\begin{array}{l}y=3x-1\hfill \\ y=3\left(0\right)-1\hfill \\ y=-1\hfill \\ \left(0,-1\right)\phantom{\rule{3em}{0ex}}y\text{−intercept}\hfill \end{array}$

We can confirm that our results make sense by observing a graph of the equation as in [link] . Notice that the graph crosses the axes where we predicted it would.

## Given an equation, find the intercepts.

1. Find the x -intercept by setting $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ and solving for $\text{\hspace{0.17em}}x.$
2. Find the y- intercept by setting $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ and solving for $\text{\hspace{0.17em}}y.$

## Finding the intercepts of the given equation

Find the intercepts of the equation $\text{\hspace{0.17em}}y=-3x-4.\text{\hspace{0.17em}}$ Then sketch the graph using only the intercepts.

Set $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ to find the x- intercept.

$\begin{array}{l}\phantom{\rule{1em}{0ex}}y=-3x-4\hfill \\ \phantom{\rule{1em}{0ex}}0=-3x-4\hfill \\ \phantom{\rule{1em}{0ex}}4=-3x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)\phantom{\rule{3em}{0ex}}x\text{−intercept}\hfill \end{array}$

Set $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to find the y- intercept.

$\begin{array}{l}y=-3x-4\hfill \\ y=-3\left(0\right)-4\hfill \\ y=-4\hfill \\ \left(0,-4\right)\phantom{\rule{3.5em}{0ex}}y\text{−intercept}\hfill \end{array}$

Plot both points, and draw a line passing through them as in [link] .

Find the intercepts of the equation and sketch the graph: $\text{\hspace{0.17em}}y=-\frac{3}{4}x+3.$

x -intercept is $\text{\hspace{0.17em}}\left(4,0\right);$ y- intercept is $\text{\hspace{0.17em}}\left(0,3\right).$

## Using the distance formula

Derived from the Pythagorean Theorem , the distance formula    is used to find the distance between two points in the plane. The Pythagorean Theorem, $\text{\hspace{0.17em}}{a}^{2}+{b}^{2}={c}^{2},$ is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse. See [link] .

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