# 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] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Sin(A+B) = sinBcosA+cosBsinA
Prove it
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