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

Try It

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

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

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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 $y=\_\_\_\_\_.$ 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 $y=2x-20$ 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 $\mathrm{-10}\le x\le 10,$ and $\mathrm{-10}\le y\le 10.$ 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: $y=-\frac{2}{3}x-\frac{4}{3}.$

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] .

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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 $y=3x-1.$

To find the x- intercept, set $y=0.$

To find the y- intercept, set $x=0.$

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.

Finding the intercepts of the given equation

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

Set $y=0$ to find the x- intercept.

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

Set $x=0$ to find the y- intercept.

$$\begin{array}{l}y=\mathrm{-3}x-4\hfill \\ y=\mathrm{-3}(0)-4\hfill \\ y=\mathrm{-4}\hfill \\ (0,\mathrm{-4})y\text{−intercept}\hfill \end{array}$$

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

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Try It

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

x -intercept is $\left(4,0\right);$ y- intercept is $\left(0,3\right).$

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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, $a^2+b^2=c^2,$ 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] .