Understanding Constant Rate of Change and Direct Variation from a Linear Graph | Lecture notes Linear Algebra

Unit8Day4

ConstantRateofChange

anddirectvariation

EssentialQuestion:

Whatdoesalineargraphthatisa

straightlinetellyouabouttherateof

change?

time(hr)

distance(mi)

100

200

300

400

(1,55)

(2,110)

(3,165)

(4,220)

TravelTIme

Thegraphshowtherelationship

betweentimeanddistanceofacar

traveling55milesperhour.

A.Chooseanytwopointsonthe

graphandfindtherateofchange.

B.Repeatpartawithdifferentpoints.

Whatistherateofchange

C.Whatdoyounoticeabouttherateofchangebetweenany

twopoints?

ThegraphfortheTravelTimeisastraightline.

Relationshipsthathavestraightlinegraphsarecalled

linearrelationships

.Noticethatasthetimeinhours

increaseby1,thedistanceincreasesby55.

Time(hr)

Distance(mi)

110

165

220

changeindistance

=

55

or55mph

changeintime1

Therateofchangebetweenanytwopointsinalinear

relationshipisthesameorconstant.Wedescribethisasa

constantrateofchange

ConstantRateofChange

NotaConstant

RateofChange

Somelinearrelationshipsareproportional.Iftheratiosareequal,

thelinearrelationshipisproportional.Iftheratiosarenotequal,

thelinearrelationshipisnonproportional.

numberofpeoplex

costofparkingy

Costofparkingy







12





Numberofpeoplex1234

Theratiosareequal,sothisisproportional.



=4

Costofticketsy







31





Numberofpeoplex1234

Theratiosarenotequal,sothisisnonproportional.

=13

=11

=11.3

=10

Numberofpeoplex

Costofticketsy

Aspecialtypeoflinearequationthatdescribesconstantrateof

changeiscalleda

directvariation

.Thisgraphalwayspasses

throughtheorigin(0,0)andisaproportionalrelationship.

The

constantofvariation

is

intheequation

y=kx

.Youcan

find

by

k=.

Youcanuseanypointontheline,oronatable,

orfromawordproblemtofindthevalueof

k





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Unit 8 Day 4

Constant Rate of Change

and direct variation

Essential Question:

What does a linear graph that is a

straight line tell you about the rate of

change?

y x 1 2 3 456 time (hr) distance (mi) 100 200 300 400 (1, 55)^ (2,^ 110) (3, (4,165)^ 220) Travel TIme The graph show the relationship between time and distance of a car traveling 55 miles per hour. A. Choose any two points on the graph and find the rate of change. B. Repeat part a with different points. What is the rate of change C. What do you notice about the rate of change between any two points? The graph for the Travel Time is a straight line. Relationships that have straightline graphs are called linear relationships. Notice that as the time in hours increase by 1, the distance increases by 55. Time (hr) 0 1 2 3 4 Distance (mi) 0 55 110 165 220 change change in distancein time = 551 or 55 mph The rate of change between any two points in a linear relationship is the same or constant. We describe this as a constant rate of change. Constant Rate of Change Constant Rate of Change Not Rate^ a ofConstant Change Some linear relationships are proportional. If the ratios are equal, the linear relationship is proportional. If the ratios are not equal, the linear relationship is nonproportional. number of people x (^1 2 3 ) cost of parking y (^4 8 12 ) Number^ Cost^ of of^ parking people^ yx^41 82 123
The ratios are equal, so this is proportional. = 4 = 4 = 4 = 4 Cost of tickets y 13 22 31 40 Number of people x 1 2 3 4 The ratios are not equal, so this is nonproportional. Number of people x (^1 2 3 4) = 13 = (^11) =11.3 = Cost of tickets y (^13 22 31 )

A special type of linear equation that describes constant rate of

change is called a direct variation. This graph always passes

through the origin (0, 0) and is a proportional relationship.

The constant of variation is k in the equation y = kx. You can

find k by k =. You can use any point on the line, or on a table,

or from a word problem to find the value of k.

y x

Let's try find the direct variation in each situation:

information (mb) time (s) x y 2.5 10 3.75 15 10 40

Understanding Constant Rate of Change and Direct Variation from a Linear Graph, Lecture notes of Linear Algebra

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Partial preview of the text

Download Understanding Constant Rate of Change and Direct Variation from a Linear Graph and more Lecture notes Linear Algebra in PDF only on Docsity!

Unit 8 Day 4

Constant Rate of Change

and direct variation

Essential Question:

What does a linear graph that is a

straight line tell you about the rate of

change?

A special type of linear equation that describes constant rate of

change is called a direct variation. This graph always passes

through the origin (0, 0) and is a proportional relationship.

The constant of variation is k in the equation y = kx. You can

find k by k =. You can use any point on the line, or on a table,

or from a word problem to find the value of k.

Let's try find the direct variation in each situation:

y = kx k = y k =

x

replace k with. y = x

The cost of cheese varies directly with the number of pounds

bought. Suppose 2 pounds of cheese cost $8.40. Write an

equation that could be used to find the unit cost of cheese.

y = kx k =

Find the cost of 3.5 pounds of cheese.

y

x