TheEquation
of a Linein Slope
y-lnterceptForm:
V: mx+ b
In Chapter2: Relations,you
discoveredthat when a person
walks at a constant speed in
front of a motion sensor,a
straight line is produced. What
information can this line tell
you about the person'smotion?
How can you use algebrato
describeand analyse a
distance-timegraph?
Investigate
,,.j
i
toots "
r Tl-84orTl-83+graphing
calculator
r CBRmmotionsensor
r maskingtape
r metrestick
r stopwatchor watch that
secon0s
measures
How canyou use an equationto describea person'smotion?
1. Find a clear location where you have plenty of room to move,
such as a hallway. Carefully measure 1-m intervals and mark with
masking tape, up to 6 m.
2. Set up for data collection:
a) One person will be the walker. Have the walker practise
walking at a constant speed of 1 m/s. Use a stopwatch to
practise. Then, get ready at the 1-m mark.
b) Connect the CBRrM to the calculator. Turn the calculator on
and load the Ranger program,
. Press@.
. Select2:CBL/CBR.
. press@.
o Select3:RANGER.
. Press@.
_LsfSBL_!tor
REALTIilE: YES
T I i E ( S lr I 5
DISPLAYI OIST
B E G I H0 n : I E N T E R ]
Sn00THInG: noaE
UNITS: I.IETERS
o Select 1:SETUP/SAMPLE.
Make sure that your settingsmatch those shown here. With
these settings,the CBRrMwill record distancesin metres for a
period of 15 s. Move the cursor up to START NOW at the top of
the screen.
Set the CBRrMso that it is 1 m from, and pointing toward, the
walker.
2 9 6 M H R . C h a p t e r6
Have
3. Press6ffi).You should hear a ticking sound from the CBRTM.
the walker start walking away from the CBRrMat a slow steady
pace of 1 m/s. A graph will begin to form on the calculator screen.
4. a) Use the blue cursor keys to trace along the graph. You will see
the coordinatesof each data point appear at the bottom of the
screen.
b) The left-most point is the y-intercept, usually labelled with the
Ietter b. Write down the coordinatesof this point. What do you
notice about this point? Why might this value not be exactly
what you expect?
TechnologYrtlP
practice
., YoumaVneeda few
.-atternptsto get a feel for
'
the conectspeed.Tryto get
. a graphthat is fairlylinear.lf
youareunsatisfied
with
yourgraph,youcancollecta
newsetof dataafterthe
ticking:
CBR'"hasstopped
.press@.
SAMPLE,
Select3:R€PEAT
c) Pick any two points along the linear part of the graph (try to use
two points that are not close together)and write down their
coordinates.Use these points to calculate the slope of the line
'-! '-1.
using the slope formula m :
Comparethe slope to the
Xz-xt
speed of the walker and describewhat you notice.
5. a) Substitute the values of b and m that you found in step 4 into
the equationy: mx * b.
bl This equation describeswhere the walker is, y, at any time, x.
Tracealong the graph and pick any point to test the equation.
o Substitutethe value of the x-coordinate into the equation you
found in part a).
r Calculate the value of y.
o Comparethis to the actual y-coordinate of that point.
r Explain why these points may not be exactly the same.
c) Repeat part b) for another point on the graph.
5. a) Now, collect a new set of motion data.
o PressG,rrii)and select 3:R-EPEATSAMPLE.
o Have your partner pick a different starting point and walking
speed (take note of thesevalues),
o Press6rei) and have your partner begin walking.
b) Repeatsteps 4 and 5 for your new graph.
7. Reflect Describehow an equation of the form y : mx ! b can be
used to describea person'smotion. What do the values of m and b
describein this case?
Form:y= mx + b' MHR 297
6.1TheEquationof a Linein Slopey-lntercept
€xample1 ldentifythe Slopeandthe y-lntercept
Identify the slope and the y-intercept of each linear relation.
Use thesevaluesto write the equationof the line.
v
v
2
U
4'
-2
- 2 0
zx
Solution
To find the slope, use two points on the graph.
a) ^ :
Yz
-
Yt
Xz- xt
1 - (-5)
Applytheslopeformula.
v
o
6 - 0
-1 +5
-2
Takecare with operations
with integers.
6
,4
:t
6
2
4
Express
fractionanswersin lowestterms.
:J
Theslopeis
l
From the graph, you can seethat the y-intercept is -5.
a
So, m :
: and b :
3
-5. Substitute these values into the
equationy=mx*b.
2
,v : 3 x * {
5 l
The equation of the line is V :f,"
2 9 8 M H R . C h a p t e r6
- s.
x
-?
-
?
b-) m :
3 - 0
_ - 6
3
:-2
The slopeis -2.
The y-interceptis 3. So, m : -2 and b : 3.
Y=-2x+3
The equation of the line is y : -2x + 3.
A
_ A
a
a
c) m : 2 - O
v
:q
l r i'-i-4.
2
: 0
-2 to
2^
The slope is zero,
The y-intercept is 4.
line.lt hasno rise.
Thisis a horizontal
Y:Ox+4
The equation of the line is
/'zeto
rise
m:run
This equation means that for any
point on this line, the y-coordinate
is always 4, no matter what the
x-coordinate is.
Zerodividedby anyvaluefor the rungiveszero,
lineis zero.
So,the slopeof anyhorizontal
d- l m : 1 - o
0 - 0
: !
0
Division
byzerogives
anundefined
result.
The slope is undefined.
There is no y-intercept.
A vertical line cannot be expressedin the form
y:mx*b.
vz
;0_
?4
{l
I ,
Look at the graph of this line. What do you notice about its
x-intercept ?
The equation for this line is x : 2.
This means that for any point on this line, the x-coordinate is
always 2, no matter what the y-coordinate is.
x-intercept
r the x-coordinate
of the
pointwherea linecrosses
the x-axis
Form:y= mx + b ' MHR 299
of a Linein Slopey-lntercept
6.1TheEquation
Example2 Grapha Line,Givenm and0
The slope and the y-intercept are given. In each case,write the
equation of the line and graph the line.
.l-:X
-2
c)m:-),,u:u
-3,b:0
d)rr1:0,b:g
Solution
a) Substitute the slope and the y-intercept into the equation
Y:mx+b.
?
v: :x + (-21
The equationof the line is V : 1* - z.
"
4
To graph this line, begin by plotting
the y-intercept, (0, -2).
Then, use the slope to locate other
points on the line.
I
Startingat (0, - 2),
;.
I move4 to the rightandup 3 to find
anotherpointon the line.I canrepeat
thisto findotherpointson the line,
Theslopeis
Once you have located two or
three points, draw and label
the line.
bly: -3 + o
-3x
Y:
The y-intercept is 0.
Start at the origin, and
-3
use m to find
1.
other points on the line.
3OOMHR. Chapter6
)
Usingthe slopein the
-t
helpsmeto
f
thinkof "rising-3"
(oractually"falling3")
1."
andthen"running
form
x
7
cly:-rx+5
The y-intercept is 5. The slope is
_!
2
1
Method
t: t-etm : t'.
Method2: Lettn:=.
Start at (0, s). Go down 1
and right 2 to find other
points on the line.
Start at (0, s). Go up 1 and
left z to find other points
on the line.
Compare these two methods. Notice that they produce
the same line.
d) The slope is zero. This is a horizontal
Iine with equationy: 6.
All points on this line
have a y-coordinate of o.
)
Example3 lnterpreta LinearRelation
Identify the slope and the vertical intercept of each linear relation
and explain what they mean. Write an equation to describethe relationship.
a)
b)
6.1TheEquationof a Linein Slopey-lnterceptForm:y- mx + b ' MHR 301
Solution
a)
Tracy'sWalk
c t 4
E
r
t ! u 2
i5
1
10'r
0
Time(s)
-?
srope:
,
.+
: -0,5
This is a distance-timegraph. The d-intercept is 5, which means
that Tracy beganwalking at a distance of s m from the sensor.
The slope is -0.5, which representsthe changein distanceover
the changein time. The negativevalue means that the distance is
decreasing.This means that Tracy'sspeed was 0.5 m/s toward
the sensor.
The equationdescribingthis relationshipis d : -0.5f + 5.
Heightol
TomatoPlant
After Planting
15
14
E 1Z
10
s tt
eo
rt, 6
E.
,2
o
1 ? 3 4 5
Time{weeksl
4
slope
1.
-
+
This is a graph of the height of a tomato plant versus time. The
lr-intercept is 6, which means that the tomato plant was 6 cm tall
when it was planted in the garden.
The slope is 4, which representsthe changein height over the
changein time. This means that the tomato plant grew at a rate
of 4 cm per week.
The equation describing this relationship is rh : 4t + B.
302 MHR.Chapter
6
Key Concepts
I The equation of a line can be written in slope
y-intercept form: y : mx + b, where
o m is the slope of the line
o b is the y-intercept of the line
r A horizontal line is written in the form y:
where b is the y-intercept. The slope of a
horizontal line is zero,
b,
r A vertical line is written in the fotm x : a,
where a is the x-intercept. The slope of a
vertical line is undefined.
YourUnderstanding
Communicate
GII The equations of four lines are given:
y:6
y:-x+4
y:2x-3
Which of these represents
x:-3
a) a vertical line?
b) a horizontal line?
c) a line that slopes upward to the right?
d) a line that slopes downward to the right?
Explain each answer you chose.
GD A line has a y-intercept of 2 and a slope
' of ].
5 n*ptain how you can
use this information to graph the line.
GFr The distance-time graph for a person walking in front
of a motion sensoris shown.
a) At what distance did this person begin walking? How do
you know?
b) Was the person walking toward or away from the sensor?
Explain how you know.
c) How fast was the person walking?
d) Write an equation in the form d : mt I b to describe the
person'smotion.
tG[l Refer to Example 3, part b).
a) Does the graph to the Ieft of the l-r-axishave meaning?
b) What would this portion of the graph represent?
c) What is the significance of the ,h-intercept?
6.1TheEquationof a Linein Slopey-lnterceptForm:y- mx + b ' MHR 303
I
Practise
For help with questions 1 to 4, seeExample L.
1. Identify the slope and the
y-intercept of each line.
Organizeyour answersin
a table like the one shown.
?
a )y : 4 x I 1
b)y:-y'*t
c lY : x - z
dly:
e)y: s
ll y: -x-;
- i zx
1
Z. Find the slope and the y-interceptof each line.
b)
v
\
z
2
- 2 0 |
/
- 2 0
-2
2)
-z
I
c)
\
d)
v
v
?
?
- - 4 - ? 0
\
2
t
- 4 \
{x
?
x
:
-2
3 . Write the equation of each line in question 2.
4. Write the equation of each line. State its slope and y-intercept,
if they exist.
a)
b)
v
v
a
- ? 0
c)
z
d)
v
-?
- 2 0
?x
v
?
2
- e 0
-4
x
?
- 7 0
?
x
-2
5. The line in question 4, part d), has a special name. What is it?
3 O 4 M H R . C h a p t e r6
For help with questions 6 to 8, see Example 2.
5. The slope and the y-intercept are given. Write the equation and
graph each line.
a)
b)
c)
d)
e)
7. Statethe slope and the y-intercept of each line, if they exist.
Graph each line.
a) y: -s
b) x : 1
cl y:
Z
2
d) x :
-2,5
ConnectandApply
8. The distance-time graph of a person walking in front
of a motion sensoris shown.
ReaenlngandPrcvint
nepeseizlng
s"htine roo',
I
1,.'\
/\
,:,"".fr/ | ---**.**
I
ProblemSolvinP
I
\,o.*ulo",'nr/
a) How far from the sensor did the person begin walking?
b) How fast did the person walk?
c) Did the person walk away from or toward the sensor?Explain.
9. Sketch a distance-time graph for each walker for the first 4 s.
a) Eleanor started at a distance of 2 m and walked away from the
sensorat a constant speed of t m/s.
b) Pierre beganwalking toward the sensorat a constant speed of
0.5 m/s from an initial distance of 5 m.
c) Jessestood at a distance of 2.5 m from the sensorand dic
not move.
d) Cassandrastarted at 1 m from the sensor and walked away from
it at a constant speed of 1 m/s for 3 s. Then, she turned around
and walked, at the samespeed,toward the sensorfor 1 s.
6.1TheEquationof a Linein Slopey-lnterceptForm:y= mx + b ' MHR 305
For help with question 70, see Example 3.
10. Identify the slope and the vertical intercept of each linear relation
and explain what they represent. Write an equation
to describethe relationship.
a)
b)
I l. ChaPtet Problem Jeangrew up in this city in
Ontario. In one part of it, there is a very steep
slope. Two of the letters in the name of this city
can be found by determining the slope and the
y-intercept of the graph shown.
ReasoninS
andProvinB
neprcseitlng
ser*'\tinc
root,
I
/\
I
;""",d
1,,'\
Solvine
-Problem
]
---':neoectine
12. Yuri tries hard not to be late for class,but sometimeshe does not
quite make it on time. Classbegins at 8:30 A.M. The distance-time
graph shows his progressfrom home to school one morning.
|
\.o"ul**,nrl
Write a story about Yuri's trip to school. Include the speed,distance,
and time in your story.
306 MHR. Chapter6
| 3. Refer to question 72. How would the graph change if Yuri left
L0 min earlier?How would this chanse affect the outcome of
vour storv?
Extend
14. Two koala bears,Rocco and Bifl are playing near a
stream. Suddenly they both rcalize that it is
dinner-time and begin to race to their eucalyptus
tree home. Their distance-time graphs are shown:
E-1"
ml-
*l+++-l-i'*ll$'
Rocco starts from the stream. which is 30 m from
home. Biff is a few metres from Rocco when he
starts.Describethis race. In your description, be
sure to mention speed,distance,and time.
1 5 .The x-intercept is the x-coordinate of the point where a graph
crossesthe x-axis.
a) What is the value of the y-coordinate for any x-intercept?
Use a diagram to explain your answer.
b) Find the x-intercept of each line.
.Jz:3x-6
-2x + 5
.Y
' : 3
r 5 . Math Contest
a) Find a number that leaves a remainder of 1 when divided by 2,
a remainder of 2 when divided by 3, and a remainder of 3 when
divided by a.
b) Find at least five other numbers that satisfy the conditions in
part a).
cf Describe a pattern or formula that can be used to find more
numbers that satisfy the conditions in part a).
6.1TheEquationof a Linein Slopey-lnterceptForm:y= mx + b ' MHR 3O7