# 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] .

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin