# 11.3 Graphing with intercepts  (Page 3/4)

 Page 3 / 4

Graph using the intercepts: $y=3x.$

Graph using the intercepts: $y=-x.$

## Choose the most convenient method to graph a line

While we could graph any linear equation    by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

Equation Method
#1 $y=2x+1$ Plotting points
#2 $y=\frac{1}{2}x+3$ Plotting points
#3 $x=-7$ Vertical line
#4 $y=4$ Horizontal line
#5 $2x+y=6$ Intercepts
#6 $4x-3y=12$ Intercepts

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form $Ax+By=C$ . We substituted $y=0$ and $x=0$ to find the x- and y- intercepts, and then found a third point by choosing a value for x or y .

This leads to the following strategy for choosing the most convenient method to graph a line.

## Choose the most convenient method to graph a line.

1. If the equation has only one variable. It is a vertical or horizontal line.
• $x=a$ is a vertical line passing through the $x\text{-axis}$ at $a$
• $y=b$ is a horizontal line passing through the $y\text{-axis}$ at $b.$
2. If $y$ is isolated on one side of the equation. Graph by plotting points.
• Choose any three values for $x$ and then solve for the corresponding $y\text{-}$ values.
3. If the equation is of the form $Ax+By=C,$ find the intercepts.
• Find the $x\text{-}$ and $y\text{-}$ intercepts and then a third point.

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}y=-3\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}4x-6y=12\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}y=\frac{2}{5}x-1$

## Solution

$\phantom{\rule{0.2em}{0ex}}y=-3$

This equation has only one variable, $y.$ Its graph is a horizontal line crossing the $y\text{-axis}$ at $-3.$

$\phantom{\rule{0.2em}{0ex}}4x-6y=12$

This equation is of the form $Ax+By=C.$ Find the intercepts and one more point.

$\phantom{\rule{0.2em}{0ex}}x=2$

There is only one variable, $x.$ The graph is a vertical line crossing the $x\text{-axis}$ at $2.$

$\phantom{\rule{0.2em}{0ex}}y=\frac{2}{5}x-1$

Since $y$ is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}3x+2y=12$
2. $\phantom{\rule{0.2em}{0ex}}y=4$
3. $\phantom{\rule{0.2em}{0ex}}y=\frac{1}{5}x-4$
4. $\phantom{\rule{0.2em}{0ex}}x=-7$
1. intercepts
2. horizontal line
3. plotting points
4. vertical line

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}x=6$
2. $\phantom{\rule{0.2em}{0ex}}y=-\frac{3}{4}x+1$
3. $\phantom{\rule{0.2em}{0ex}}y=-8$
4. $\phantom{\rule{0.2em}{0ex}}4x-3y=-1$

1. vertical line
2. plotting points
3. horizontal line
4. intercepts

## Key concepts

• Intercepts
• The x- intercept is the point, $\left(a,0\right)$ , where the graph crosses the x- axis. The x- intercept occurs when y is zero.
• The y- intercept is the point, $\left(0,b\right)$ , where the graph crosses the y- axis. The y- intercept occurs when y is zero.
• The x- intercept occurs when y is zero.
• The y- intercept occurs when x is zero.
• Find the x and y intercepts from the equation of a line
• To find the x- intercept of the line, let $y=0$ and solve for x .
• To find the y- intercept of the line, let $x=0$ and solve for y .
x y
0
0
• Graph a line using the intercepts
1. Find the x- and y- intercepts of the line.
• Let $y=0$ and solve for x.
• Let $x=0$ and solve for y.
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.
• Choose the most convenient method to graph a line
1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
$x=a$ is a vertical line passing through the x- axis at a .
$y=b$ is a vertical line passing through the y- axis at b .
2. Determine if y is isolated on one side of the equation. The graph by plotting points.
Choose any three values for x and then solve for the corresponding y- values.
3. Determine if the equation is of the form $Ax+By=C$ , find the intercepts.
Find the x- and y- intercepts and then a third point.

## Practice makes perfect

Identify the Intercepts on a Graph

In the following exercises, find the $x\text{-}$ and $y\text{-}$ intercepts.

(3,0),(0,3)

(5,0),(0,−5)

(−2,0),(0,−2)

(−1,0),(0,1)

(0,0)

Find the $x$ and $y$ Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

$x+y=4$

(4,0),(0,4)

$x+y=3$

$x+y=-2$

(−2,0),(0,−2)

$x+y=-5$

$x-y=5$

(5,0),(0,−5)

$x-y=1$

$x-y=-3$

(−3,0),(0,3)

$x-y=-4$

$x+2y=8$

(8,0),(0,4)

$x+2y=10$

$3x+y=6$

(2,0),(0,6)

$3x+y=9$

$x-3y=12$

(12,0),(0,−4)

$x-2y=8$

$4x-y=8$

(2,0),(0,−8)

$5x-y=5$

$2x+5y=10$

(5,0),(0,2)

$2x+3y=6$

$3x-2y=12$

(4,0),(0,−6)

$3x-5y=30$

$y=\frac{1}{3}x-1$

(3,0),(0,−1)

$y=\frac{1}{4}x-1$

$y=\frac{1}{5}x+2$

(−10,0),(0,2)

$y=\frac{1}{3}x+4$

$y=3x$

(0,0)

$y=-2x$

$y=-4x$

(0,0)

$y=5x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$-x+5y=10$

$-x+4y=8$

$x+2y=4$

$x+2y=6$

$x+y=2$

$x+y=5$

$x+y=-3$

$x+y=-1$

$x-y=1$

$x-y=2$

$x-y=-4$

$x-y=-3$

$4x+y=4$

$3x+y=3$

$3x-y=-6$

$2x-y=-8$

$2x+4y=12$

$3x+2y=12$

$3x-2y=6$

$5x-2y=10$

$2x-5y=-20$

$3x-4y=-12$

$y=-2x$

$y=-4x$

$y=x$

$y=3x$

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

$x=2$

vertical line

$y=4$

$y=5$

horizontal line

$x=-3$

$y=-3x+4$

plotting points

$y=-5x+2$

$x-y=5$

intercepts

$x-y=1$

$y=\frac{2}{3}x-1$

plotting points

$y=\frac{4}{5}x-3$

$y=-3$

horizontal line

$y=-1$

$3x-2y=-12$

intercepts

$2x-5y=-10$

$y=-\frac{1}{4}x+3$

plotting points

$y=-\frac{1}{3}x+5$

## Everyday math

Road trip Damien is driving from Chicago to Denver, a distance of $1,000$ miles. The $x\text{-axis}$ on the graph below shows the time in hours since Damien left Chicago. The $y\text{-axis}$ represents the distance he has left to drive.

Find the $x\text{-}$ and $y\text{-}$ intercepts

Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Damien.

(0,1,000),(15,0). At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive.

Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The $x\text{-axis}$ on the graph shows the number of miles Ozzie drove since filling up. The $y\text{-axis}$ represents the number of gallons of gas in the truck’s gas tank.

Find the $x\text{-}$ and $y\text{-}$ intercepts.

Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Ozzie.

## Writing exercises

How do you find the $x\text{-intercept}$ of the graph of $3x-2y=6?$

Answers will vary.

How do you find the $y\text{-intercept}$ of the graph of $5x-y=10?$

Do you prefer to graph the equation $4x+y=-4$ by plotting points or intercepts? Why?

Answers will vary.

Do you prefer to graph the equation $y=\frac{2}{3}x-2$ by plotting points or intercepts? Why?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Prealgebra' conversation and receive update notifications?

 By By Mistry Bhavesh