Algebra Tutorial-linear Graphs

Algebra Tutorial Linear Equations Striaght lines are Most of the data

easy to construct and use. from real world are first fitted to straight line graphs

Consider this. If the price of a commodity increases, ususally its demand or sales drop. We shall write a simple equation for this "linear Model": Sales is denoted by S S=

Price is denoted by P a-

bP

It is of the same form as the straight line equation:

Y=A-BX

We know that for any straight line, we put in two 'constants' in the equation.Here the constants are:

and

A ,intercept B , the slope

For the given problem,let us take specific example: Sales figures of various cars follow this equation: Sales per week S = 100000 100000- 0.1 - 8XP 0x xPP where P is given in thousands of dollars. For example, if the cost of the car is $ 20000 , Sales per week S = 100000 - ).180x20 x 20 98400 In this equation , intercept = 100000 and slope is 80. Let us plot this equation:

To draw a graph, we need points on the straight line.We ge forming a small table.This is always a useful practice. P

S 0 100000 10 99200 20 98400 30 97600 We have four points on the straight line. Note: we can draw a straight line with only tow points.But it to check th eline.With two point ,if we make mistake incalcu Now let us draw this line using Excel spreadsheet program. S 0 10 20 30

100000 99200 98400 97600

sales

P

Sales versus Price

100000 99750 99500 99250 99000 98750 98500 98250 98000 97750 97500 0

2.5

5

7.5

10

12.5

15 Price

17.5

20

22.5

sales

99750 99500 99250 99000 98750 98500 98250 98000 97750 97500 0

2.5

5

7.5

10

12.5

15

17.5

20

22.5

Price

Note that the inrecept at price =0 is 100000. This ahs mathematical meaning.It has no physical meaning there is no sales to speak of---unless we are including charity and gifts---well that is not business!

What is slope? It is just the RATE of drop in slaes when price goes up bu one unit--in this case,a thousa When price incresease from 10 to 20 thousand dollars,sales drops from 99200 to 98400. SLOPE

800 /10 80 This is what we have put inth eequation: S = 100000 - 80xP

This line has negative slope: As P increases

S decreases.

The line goes downhill.

Line with positive slope

Let us see an example. John finds that the weight of his son , James, is increasing from the bir Let us set up an equation and plot the graph: The equation would be : weight W = 8 + 0.5 T where W is the weight of the baby-boy in Lbs and T is the time in months. Here in mathematical terms, intercept a = 8 , the birth weight ar the weight at time T =0 The slope is the rate of increase which is b = 0.5 ; this slope is positive. As T increases, W

Let us pick a few points on the stright-line to plot the graph". You can do this with a graph paper or quadrille paper.I will show with the help of Excel spreadsheet prog Weight versus Age 11 10.5 Wieght (lbs)

T (months) W (lbs) 0 8 2 9 4 10 6 11

10 9.5 9 8.5 8 0

1

2

3

4

5

Age (months)

It is important to label the two axes-- X and Y with the quantities markes---here age and weight.You can

Line with no slope or zero slope. Suppose I write: y = 3. then this value does not vary with X…so the line is astraight line horizontal,parall [ A line x =2 which is a vertical line has 'infinite' slope oe it is undefined . Let us plot y = 3

X

Y

Y = 3 graph

3 3 3 3 3

y

0 2 4 6 10

3 2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 0

1

2

3

4

5

6

7

8

x

Study the two examples of line with postive slope and negative slope. Line with zero intercept You can have line with zero intercept: y = b x When x =0, y is also zero…Therefore the origin (0,0) is a point on this graph. The straight line passes through the origin. What could be simple than this.

Let us see an example. Bill sells ice cream cones at the rate of $1.99 for each cone.His revenue depen Revenue R = 1.99 x N $ Here the slope is positive and equal to N. Revenue versus Sales Let us plot this equation: 8 R 0 1 2 4

0 1.99 3.98 7.96

Revenue ($)

7

N

6 5 4 3 2 1 0 0

0.5

1

1.5

2

2.5

3

N (# sold)

Application Problems 1 A boat goes at the constant speed of 30 miles per hour. If the boat is at a distance of 20 miles from the Distance from the shore D = 20 + 30 x T where T is the time in hours. This is a straight line graph of distance versus time. {Time in the X axix and distance Y axis. 2 A glucose-control drug is given to a patient which decreases in effect at the rate of 5 % every hour. Draw the effect of drug which is 100% at time t=0 with passage of time. Effect = 100 - 5 x t where t is in hours.

3 Amanda bakes ckaes and sells them at a local store. She has spent $200 for equipment needed for he

Set up the equations for her cost and revune and plot and analyse the results as to the profit/loss she wi Cost equation: C= 200 + 0.3 x N where N is the number of cakes produced Revenue R = 1.5xN We can plot bothe the straight lines ina chart and analyse. C 0 100 200 400

R 200 230 260 320

Revenue-Cost Relation

0 150 300 600

600 500 Cost or Revenue

N

400 300 200 100 0 0

25

50

75 100 125 150 175 200 225

N Number mad

Business Analysis

From the graph, we see that the cost line and revenue line cut each other at N= 1 If Amanada is able to sell more than 167 cakes ,she will make profit.In fact she be Profit or loss = R - C = 1.5 N - (200 + 0.3 N) This line represents profit or loss versus N,number sold. Plot this also in the same graph for practice.

To Sum Up: We have written equations of straight lines with positive slope,negative slope and also equations with no We have seen several examples to set up such equatuions and to extract useful information for decision The business example illustrates a practical apporach using graphs.

ere the constants are:

s on the straight line.We generate a few points by ays a useful practice.

e with only tow points.But it is better to pick four points if we make mistake incalculation,the straight line will be wrong.

ales versus Price

S

12.5

15 Price

17.5

20

22.5

25

27.5

30

S

12.5

15

17.5

20

22.5

25

27.5

30

Price

It has no physical meaning because when price is zero,, there cannot be any sales as that is not business!

e unit--in this case,a thousand dollars in price.

The line goes downhill.

s, is increasing from the birth weight of 8 Lbs at the rate of ).5 lbs every moth. Plot the graph and predict Jmases weight when Janmes rea

e weight at time T =0 ositive. As T increases, W increases too.

p of Excel spreadsheet program here. Weight versus Age

W (lbs)

2

3

4

5

6

Age (months)

e age and weight.You can also write the units inbrackets as I have shown…These are 'good habits' to learn.!

traight line horizontal,parallel to X axis.

Y = 3 graph

Y

4

5

6

7

8

9

10

x

ch cone.His revenue depends on the number of cones sold.Let n be the number of cones he sells.What is his revenue in mathematical term Revenue versus Sales

R

1

1.5

2

2.5

3

3.5

4

N (# sold)

stance of 20 miles from the shore, write the equation for the distance at any given time:

istance Y axis. rate of 5 % every hour.

or equipment needed for her kitchen.The material and heating costs for making one cake is roughly 30 cents. She sells at the rate of $1.50 f

as to the profit/loss she will get.

Revenue-Cost Relation

75 100 125 150 175 200 225 250 275 300 325 350 375 400 N Number made

line cut each other at N= 166 cakes.What does this mean? ll make profit.In fact she begins to get profit onle after that sales figure.If she sell less, she suffers a loss. Beyond this point,called "break -ev

s versus N,number sold.

and also equations with no intercept. eful information for decision making.

s weight when Janmes reaches tow years.

enue in mathematical terms?

e sells at the rate of $1.50 for each cake to the store.If she

this point,called "break -even point", the revenue curve is above the cost curve .below this point,cost line is above revenue line.

ve revenue line.