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

 Page 3 / 21

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] .

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice