Unit 3: Linear Relations - Mr. McDonald's - Mr....

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Unit 3: Linear Relations

Prior Knowledge

Slope or m = rise run

1. Calculate the slope of a line that has a rise of 15m and a run of 6 m.

2. If a line has a slope of 2.5 and it's run is 9 m, what will be its total rise?

Math 3202 Unit 3: Linear Relations Handouts

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m = y2 y1 x2 x1

Example: Calculate slope using 2 points:

(2, 7) (4, 11) (2, $25.00) (5, $62.50)

Example:

1. Calculate the slope of the following:

(7, 13) (9, 5) (2, $115) (5, $220)


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0 1 2 3 4 5

1

2

3

4

5

x

y

Independent Variable

Dependent Variable

The independent variable is the one that is changed. (eg) you can change the number of shoes you buy. Time is another example of an independent variable.

The dependent variable is the one that is effected by the change to the independent variable (eg) the number of shoes you buy will affect the total cost. (eg) Number of Shoes

(eg) Total Cost ($)

Sec 3.1 Linear Relations in Tables and Graphs

A linear relation is a relationship between two variables that forms a straight line when you plot values of ordered pairs on a graph.

(eg) 12 inches = 1 foot, then 36 inches = 3 feet

50 songs = 8 Gbits of storage space, then 200 songs = 32 GB

1 slurpie = $2, then 4 slurpies = $8


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0 2 4 6 8

4

8

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Cost

($)

Number of Slurpies

Cost of Slurpies compared to Number of Slurpies

If you were to plot a graph comparing the number of slurpies to cost, you would notice the following characteristics.

* The graph is a straight line.* It contains the ordered pair (0,0), called the origin.* The ratio of Slurpies to cost is constant.* The Slope of the graphed line is also constant.

This is called a Direct Linear Relation

Example Julio is installing an automatic watering system in a small greenhouse. He needs to measure the maximum flow rate of water from the water supply. He has a rain barrel that he knows holds 340 litres of water. He empties the barrel, turns on the water supply fully, and notes that it takes 16 minutes and 30 seconds to fill the barrel completely.

1. Do you expect the relationship between the volume of water in the barrel andthe elapsed time to be linear? Why or why not?


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2. How many data points are provided? Are there enough to draw agraph?

3. Graph the volume of water in litres compared to time in seconds.

Example: Julio has a rain barrel that he knows holds 340 litres of water. He empties the barrel, turns on the water supply fully, and notes that it takes 16 minutes and 30 seconds to fill the barrel completely.

4. Use the formula for slope to calculate the slope of the line. Which isthe independent variable and which is the dependent variable?

5. What are the units of the rise? The run? How can this slope valuebe expressed as a rate of change? (Hint: Consider the units used to measure the variables.)

6. What is the flow rate, in litres/second, of the water supply?

7. Look at the graph and estimate how many litres of water were in thebarrel after 5 minutes and after minutes.


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Partial Linear RelationsThe graphs you have examined so far in this chapter have had three characteristics in common.

* They have all been straight lines.

* They have all had positive slopes.

* They have all included the ordered pair (0, 0)

They have all been direct linear, positive relationships. Many relationships between variables, however, do not share all three of these characteristics. When a linear relation does not pass through the origin (0,0) it is a partial linear relation.

Example

Anjali is an events coordinator at a banquet hall. She is arranging a wedding reception for clients. They have not yet decided on the number of quests, so they ask Anjali what the price would be for between 50 and 200 people. Anjali creates a graph to show the prices.

a) Is the relation between the number of guests and cost linear?Explain why.

b) Does the line pass through the origin (0,0)? Why or why not?

c) What is the slope of the line? What rate of change does it represent?


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Example

Classify each of the following graphs according to the following criteria.

a) Is the graph of the relationship linear or nonlinear?

b) Is the depend variable increasing or decreasing as the independent variableincreases?

c) Do you agree or disagree with the statement, "Graph 3 passes through the originand is therefore a direct linear relation"? Justify your response.


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1. Marian is a salesperson in an electronics store. She earns a basesalary of $1280.00 per month, and she earns a commission of 3% on all her sales.

a) Create a table of values showing Marian's earnings if she makessales of the following amounts: $500.00, $1000.00, $1500.00, $2000.00 and $2500.00.

b) Draw a graph that displays the data in the table of values.

c) Describe the graph. Is the relation linear? Does the graph passthrough the origin? Include your reasons.

d) Calculate the slope of the graph. What rates does the sloperepresent?

Assignment Section 3.1 Name: ____________


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2. Shane wants to join a local tennis club. The club charges an initialfee of $95.00 to join, then a monthly fee of $20.00

a) Create a graph that shows how much Shane will spend in totalcompared to the number of months of membership, for up to 12 months.

b) Describe the shape of the graph.

c) Calculate the slope of the graph. What does this value represent?


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3. The volume of a cylinder in a gasoline engine is defined by itsdiameter (bore) and by the distance that the piston travels on each stroke (stroke length).

Graph the relationship in the table of values and verify whether it represents a linear or nonlinear relation. Would you expect the plot to pass through the point (0,0)? Why does the data not show this point?


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Example 1

The following table of values shows the relationship between air temperature and elevation above sea level under certain atmospheric conditions.

Elevation above sea level (m) Air temperature (0 C)

0 0

100 1

200 2

300 3

400 4

Air Temperature Comparedto Elevation Above Sea Level

a) Graph the data

Elevation above sea level (m)

Air temperature (0 C)

0 0

100 1

200 2

300 3

400 4

Air Temperature Comparedto Elevation Above Sea Level

b) Is the relationship between elevation and air temperature linear?Is it a direct linear relation?

Sec 3.2 The Equation of a Line

Recall the equation for calculating the slope of a line segment on graph.

m = y2 y1 x2 x1


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c) Calculate the slope of the line.Express the rate of change, using the units for each variable.

d) Write an equation that describes therelation between air temperature and height above sea level.

e) Use this equation to calculate the air temperature at the followingelevations:i) 239 metersii) 520 meters

f) Use this equation to calculate the elevation at which the following airtemperatures are observed.

i) 3.7o C ii) 6.8o C


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In a partial linear relation, the graph forms a straight line, but it does NOT pass through the origin. The point at which the line crosses the yaxis is called the yintercept.

A partial linear relation can be represented as the following equation, where b is the yintercept. Remember that m is always the slope!

Let's Review

y = mx +b

ExampleTianna has a contract with the town council to operate a wading pool in the community park during the summer months. As a safety precaution, the pool must be emptied every afternoon by 4:00 pm, when it closes. Tianna needs to know how long it takes to empty the pool so that she can start the pump far enough ahead of time, but still get the maximum water play time.

On the first day the pool is open, she starts the pump at 2:30 pm. At this time, she measures the depth of the water to be 32 cm. Fortyfive minutes later, she measures the depth again and it is 11 cm. Will the pool be empty before 4:00 pm? When should she start the pump in the future so that the pool is empty at exactly 4:00 pm?


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Example Charlie runs a bed and breakfast in Dawson City, Yukon. He has two thermometers outside: one records degrees Celsius, the other degrees Fahrenheit. In one year, he records the maximum summertime temperature of 32.60 C and 90.680 F. He records the minimum winter temperature in that year as 460 C and 50.80 F.

a) Write a linear equation that describes the relationship betweendegrees Celsius and degrees Fahrenheit.

b) What is the temperature in degrees Fahrenheit when it is 00C?

c) What is the temperature in 0C, when it is 750 F


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Example Annika is deciding whether to join a fitness club or use a local community centre. The community centre charges a fee of $8.00 per use. The fitness club has a $700.00 yearly membership fee, but then charges only $2.00 per use.

a) Write equations to show the yearly cost for each of the two options.

b) If Annika used the facility twice a week all year, which option would

be less expensive?

c) How many times would Annika need to use one facility for the costto be the same for both facitilies?


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Assignment: Section 3.2

Name: ___________________

1. Sue is comparing two monthly phone plans. Company A charges aconstant rate of $0.35/minute. Company B charges a monthly fee of $12.00 plus $0.18/minute.

a) Write an equation describing each company's phone plan.

b) Sue estimates that she used 60 minutes of phone time each month.Which plan should she choose?

2. Susanna is a veterinary technician. When administering a liquidantibiotic to dogs, she must give an animal 3 ml of the medicine for each kilogram of its weight.

a) Write an equation that describes the relation between the amount ofantibiotic administered (in mL) and the weight of the dog (in kilograms).

b) What amount of antibiotic should Susanna administer to a dog thatweighs 27 kg?


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Sec 3.3 Scatterplots and Linear TrendsA scatterplot is a graph of ordered pairs, used to evaluate what type of relationship (if any) exists between two variables.

The scatterplot above graphs the relationship between turbidity and total suspended solids. While a straight line cannot be drawn to pass through all the data points, one can be draw to pass through the middle of most of the data points, to show the general trend; this is called a line of best fit.

If the line of best fit passes though the origin, a direct linear relation exists.

If it does not pass through the origin, the line describes a partial linear relation.

The line of best fit on a scatterplot provides the following information about the relationship between two variables:

* the shape (linear or nonlinear)

* the strength (weak or strong)

* the direction (positive or negative)

* presence of outliers


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The strength of a relation is shown by the amount of "scatter" in a scatterplot. If the data points are widely spread but still follow a similar trend, the correlation between the variables is weak. If the data points are concentrated around the line of best fit, the correlation (or the degree of relatedness between variables) is strong.

An outlier is a data point that lies significantly outside the general trend in the data. The data point of an outlier will be far from the line of best fit compared to the other data points. There are many possible causes for an outlier to appear, including measurement errors or equipment malfunctions, and it in important to investigate their cause.

It is common practice, however, when writing equations representing trends on a scatterplots, to discard the outliers.

In the previous graph example, the data shows:

* a linear relation, because the data form a relatively straight line.* a strong correlation, because the points are very close to the line of best fit.* a positive relation, because the line of best fit slopes upward as the values onthe yaxis increase.* no outliers.

Scatterplots are especially useful with large numbers of data points, because they help you visualize trends in the data.

For scatterplots showing linear relations, you can write an equation to express one variable in terms of the other. First calculate the slope, then substitute it into the equation of a line you used in the previous section.


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Discuss the Ideas: Evaluation Correlation Between Two Variables

Examine the scatterplots below.

1. How are the lines of best fit of scatterplots 1 and 2 similar?

2. How are scatterplots 1 and 2 different?

3. Which graph shows a stronger correlation, and why?

4. Consider the scatterplot below. Describe the strength of the relationshipbetween the variables.


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Example

Mario works in the quality control department for a company that manufactures tub connector fittings for aircraft hydraulic systems. One of his tasks is to monitor the inner diameter of the connector fittings to ensure they will fit properly on the hydraulic tubing. He collects the data about undersized fittings shown on the graph below for various production runs of elbow fittings.1 "

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a) Does the relationship between the size of production run and the number of fittingswith an undersized inner diameter appear to be linear or not? Explain your thinking.

b) Should a line of best fit pass through the point (0,0)? Why or why not?


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c) Consider the correlation between the size of production run and the number of fittingswith an undersized inner diameter. Does the correlation appear weak or strong? Why?

d) Examine the data for outliers. Why might there be a high number of undersizedfittings in one production run? Should Mario include this point if he wants to determine the rejection rate?


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Example 2

The graph below compares new vehicle fuel economy and vehicle weight for city and highway driving.

a) Does the correlation between variables appear stronger for city or highway driving?

b) Is the relationship between fuel economy and vehicle weight a direct linear

relationship or a partial linear relationship?


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c) Write two equations that can be used to estimate fuel economy compared to

vehicle weight for the two driving conditions.

d) Estimate the city and highway fuel economy of a vehicle weighing 900 kg.


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1. A farmer is assessing how much fertilizer his plots of tomato plants need. He has agreenhouse that houses 100 m2 plots of tomatoes. He decides to use different quantities of fertilizer on each plot, and to measure the yield of tomatoes. He makes a table of values showing the amount of fertilizer he used, and the total yield of tomatoes.

a) Graph the data on a scatterplot. Does therelation appear to be linear? If so, draw a line of best fit.

b) Identify any outliers. What might have causedany variation in the data?

c) Write an equation that describes the tomatoyield compared to the quantity of fertilizer used.

Assignment: Section 1.3Name: ________________


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2. Katherine and Anne are sisters. Over the years, their parents track their growth onthe following growth charts

a) Write an equation that describes the line of best fit comparing Katherine's height toher age.

b) Using this equation, calculate what Katherine's height will be when she is 45 years olExplain whether or not this is a good estimate of her height at age 45 and why.

c) Write an equation that describes the line of best fit comparing Anne's height to her age.

d) Based on the current growth trends, who will be taller at age 12, Katherine or Ann


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3. Mario works in the receiving department of a store that sells dinnerware. He monitorsthe breakage rate of dinnerware that is received from the manufacturers. He begins to notice a pattern in breakage linked to the freight companies used to ship the dinnerware. He examines the receiving records and plots the data in the graph shown here.

a) For Freight Company A, does there appear to be a correlation between the size of the order and the breakage? If so, write an equation that describes the relation.

b) For Freight Company B, does there appear to be a correlation between the size of the order and the breakage? If so, write an equation that describes the relation.


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c) For Freight Company B, what is the maximum order size for which no breakagemight be expected?

d) Based on his analysis, what recommendation should Mario make to minimizebreakage?


Unit 3: Linear Relations - Mr. McDonald's - Mr....

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