CHAPTER Linear Functions 2 · 2019-02-07 · 2 CHAPTER Skill Builder: Introduction to Linear...

Skill Builder

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2CHAPTER

Skill Builder: Introduction to Linear Functions 93

Linear Functions

1 State the rule of the following function.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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8

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2

2 State the rule of the following function.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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8

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2

3 Determine if the following function has a direct or partial variation.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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4 Determine if the following function has a direct or partial variation.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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2

1 Introduction to Linear Functions EVALUATED


Skill Builder

94 Chapter 2: Linear Functions

5 State the slope, y-intercept, domain and range of the following linear function.

2 3y x= +

My Calculations

6 State the slope, y-intercept, domain and range of the following linear function.

5y x= −

My Calculations

7 Find the zero (x-intercept) of the following linear function.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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10

8

6

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2

8 Find the zero (x-intercept) of the following linear function.

3 12y x= − +

My Calculations

9 If a horizontal line passes through the point (3, 2), what is the rule for that line?

My Calculations

10 If a vertical line passes through the point (5, –7), what is the rule for that line?

My Calculations


MHS Critical Thinking and Big Ideas 95

CRITICAL THINKING AND BIG IDEAS

Introduction to Linear Functions

1. Mary is a math teacher and knows that students very often get confused by the equations of horizontal and vertical lines because they have always associated the x-axis with the horizontal and the y-axis with the vertical. Please formulate a memory association to help Mary’s students remember the equations of horizontal and vertical lines.

Equation How to Remember

Horizontal Line

Vertical Line

2. Fill in the blanks

a. The graph of a direct variation does this, but the graph of a partial variation does not. The answer is “ ”.

b. The rule of a partial variation is .

c. The rule of a direct variation is .

d. The equation of a horizontal line passing through the point (5, 3) is .

e. The equation of a vertical line passing through the point (–2, 1) is .

f. The equation of the x-axis is .

g. The equation of the y-axis is .


96 MHS Critical Thinking and Big Ideas

3. Marco has just been hired as the urban planner for a new residential development project. As part of the development’s municipal obligations, they need to provide a park for the neighbourhood. Marco has planned for a park to be situated in the rectangle created by the intersections of the roads represented by the following equations. { }10, 40, 15, 50x x y y= = = =

Graph the four lines and shade in the area for the park on the graph. Given that the graph is scaled in metres and the cost of sod is $4.75 per m2, how much will it cost for Marco to have a crew lay sod in the park?

Show all work


Skill Builder

Skill Builder: Slope of a Line 97

1 Find the slope of the following function.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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(2,-3)

(8,6)


-10 -8 -6 -4 -2 2 4 6 8 10-2

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2

(7,-6)

(2,4)


-10 -8 -6 -4 -2 2 4 6 8 10-2

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10

8

6

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2

(1,-3)

(4,6)


-10 -8 -6 -4 -2 2 4 6 8 10-2

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10

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6

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2

(-5,-6)(5,-2)

2 Slope of a Line EVALUATED

Skill Builder



5 Find the slope of the straight line between the following points. ( ) ( )1,6 and 6, 22A B− −

My Calculations

6 Find the slope of the straight line between the following points. ( ) ( )2,7 and 5, 8A B −

My Calculations

7 Show that the slope of the straight line between AB is the same as BA for the following points. ( ) ( )4,9 and 21, 91A B− −

My Calculations

8 Show that the slope of the straight line between AB is the same as BA for the following points. ( ) ( )12, 5 and 2, 23A B− −

My Calculations

9 Show that the following points are collinear by plotting the points on a graph and by finding the slope between the points. ( ) ( ) ( )6, 8 , 3, 5 and 9, 3A B C− − − −

My Calculations

10 Show that the following points are collinear by plotting the points on a graph and by finding the slope between the points. ( ) ( ) ( )8, 4 , 4, 3 and 4,1A B C− −

My Calculations




Slope of a Line

1. Demonstrate your understanding of “Slope”.

a. Write a definition for the term “slope” in words.

b. What is the formula (or equation) for calculating slope?

c. A horizontal line segment has a slope of .

d. A line segment that is increasing (going uphill from left to right) has a slope.

e. A line segment that is decreasing (going downhill from left to right) has a slope.

f. Why does a vertical line segment have an “undefined” slope?



2. Sofia is learning about slope and is confused about how to label the points as she enters them into the slope equation. Her teacher gave her a graph with the points A(2, 7) and B(6, 10) and then asked her to calculate the slope of AB and the slope of BA.

a. Help Sofia by calculating the slope of AB and the slope of BA.

b. What have you illustrated here?

c. Why is this not a “proof”?

d. What do you think is the most frequent mistake that students make when calculating slope?



3. Alfred’s, Barbara’s and Callum’s houses all lie on the same street. On the Cartesian grid, their houses can be represented by the points (1, 8), (3, ?), (7, 26)A B C . Points are collinear if a single line can be drawn that passes through all of them. Given that their three houses are collinear, explain in detail the steps necessary to determine the missing y-value for Barbara’s house.

Detailed steps

Skill Builder



1 Find the rule of the following linear function in slope – intercept form.

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(7,-8)

(1,4)

2 Find the rule of the following linear function in slope – intercept form.

-10 -8 -6 -4 -2 2 4 6 8 10-2

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8

6

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2

(9,5)

(-6,-5)

3 Find the rule of the following linear function in slope – intercept form using the given coordinates. ( ) ( )1, 2 and 4,13− − −

My Calculations

4 Find the rule of the following linear function in slope – intercept form using the given coordinates. ( ) ( )3, 5 and 6, 2−

My Calculations

5 Find the rule of the following linear function in slope-intercept form that has a slope of 2 and passes through the point (–1, 4).

My Calculations

3 Functional (Slope-Intercept) Form of a Line EVALUATED

Skill Builder


Skill Builder: Functional (Slope-Intercept) Form of a Line 103

6 Find the rule of the following linear function in slope-intercept form that has a slope of –3 and passes through the point (2, –5).

My Calculations

7 Find the rule of the following linear function in slope-intercept form that has an initial value of 4 and passes through the point (2, 5).

My Calculations

8 Find the rule of the following linear function in slope-intercept form that has an initial value of 6 and passes through the point (–2, 0).

My Calculations

9 Find the rule of the following linear function in slope – intercept form using the given coordinates. ( ) ( )2, 5 and 3, 5−

My Calculations

10 Find the rule of the following linear function in slope – intercept form using the given coordinates. ( ) ( )4,6 and 2, 3− −

My Calculations




Functional (Slope-Intercept) Form of a Line

1. What are the steps required to find the equation of a line in the form y ax b= + given two points? Provide an example and apply your two steps to determine the equation of the line.

STEP 1:

Step 2:

Example:



2. The following table represents a linear function. Find the rule of the linear function and then use it to fill in the table and answer the following questions.

x 0 2 6 8 10 11 n

y 24 36 48 52 60

Show work

a. What is the rule of this linear function in Slope-Intercept Form?

b. Would the graph of this function be increasing or decreasing?

c. How do you know the answer to part b by analyzing the equation of the line?

d. What are the coordinates of the x-intercept of this function?



3. For each of the following, determine the rule of the linear function that could be used to model the given situations below, state what each of the variables in the rule represents and answer the question.

a. Ismail observed that a plant’s growth is directly proportional to time. The plant measured 5 centimetres in height when it arrived at the nursery and 8 centimetres exactly one week later and the plant continues to grow at that same rate. He is able to sell the plant when it reaches a height of 65 centimetres, how long will he have the plant?

b. Easy Wreck Car Rentals charges $40 per day to rent a car plus an additional $0.15 per kilometre. How much would it cost to rent the car for 5 days and travel 275 kilometres?



c. The Chicoutimi Racquet Club charges a membership fee and an additional fee for each visit to the club. Claude pays $95 for 5 visits and Danielle pays $113 for 11 visits. How much would Jean pay for 20 visits?

d. Helène owns a car that depreciates by the same amount each month. The car is currently 31 months old and the trade-in value is $17,800. The trade-in value when the car was 24 months old was $19,200. What was the car worth when it was brand new?

Skill Builder



1 Find the rule of the following linear function in general form.

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2 (6,0)(0,2)

2 Find the rule of the following linear function in general form.

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(9,5)

(-6,-5)

3 Find the rule of the following linear function in general form that has a slope of 2 and passes through the point (–1, 4).

My Calculations

4 Find the intercepts of the following linear function. 3 6 12 0x y− + =

My Calculations

5 Convert the following linear function into general form.

32

5y x= −

My Calculations

4 General Form of a Line ENRICHMENT

Skill Builder


Skill Builder: General Form of a Line 109


0.7 0.34y x= +

My Calculations


13 5

yx+ =

My Calculations


( )3 4 2y x+ = −

My Calculations

9 Convert the following linear function into slope – intercept form. 3 2 8 0x y+ − =

My Calculations

10 Convert the following linear function into

symmetric form 1yx

a b+ = where a is the

x-intercept and b is the y-intercept

4 5 40 0x y− − =

Hint: this question is just testing your algebra skills and your ability to find the intercepts in General Form, not your knowledge of “Symmetric Form”. Simply calculate the x-intercept and substitute it in for a, then calculate the y-intercept and substitute it in for b.

My Calculations




General Form of a Line

1. Mathematicians like to use the general form of a Line 0Ax By C+ + = because there are no fractions or decimals to deal with. Please complete each of the following to demonstrate your understanding of the general form of a line.

a. A, B and C must be

b. The value of A must be

c. The value of the slope is equal to

d. The y-intercept is equal to

e. Explain the steps necessary to convert from slope-intercept form to general form.



2. Working with the General Form.

a. Find the rule of the linear function illustrated in the graph in General Form

RULE:

b. Find the zero (the x-intercept) of the function 8 2 24 0x y− + =



c. Determine the equation of the line that passes through the points (2, 6) and (5, 2) in the general form.

3. Complete the table by converting the functions to or from General Form.

General Form Slope-Intercept Form

− + =6 5 10 0x y

= − +0.6 0.25y x

+ − =8 27 0x y

= +4 17 2

y x

Skill Builder


Skill Builder: Graphing Linear Functions Level 1 113

5 Graphing Linear Functions Level 1 EVALUATED

1 3 4y x= −

My Calculations

2 2

63

y x−

= +

My Calculations

3 6y x= −

My Calculations

4 4y x=

My Calculations

Graph the following linear functions using the y-intercept and the slope.

Skill Builder



5 5y x= − +

My Calculations

6 3y x= +

My Calculations

7 64x

y = −

My Calculations

8 3

45

y x= − +

My Calculations

9 53x

y = − +

My Calculations

10 6y =

My Calculations




Graphing Linear Functions Level 1

1. Your friend Sandrine missed a day of school for a doctor’s appointment and has texted you for help. Please write a detailed step-by-step explanation for Sandrine and make sure to include when she should use this method.

What did I miss in math class today?

We learned how to graph lines using the y-intercept

and slope

I have no idea how to do that. Is it difficult?

It’s not bad if you know the steps and remember when it’s good to use it.

Can you help me?



2. Chad needs to drain his aquarium in order to clean it. Cleaning the tank is relatively simple, but draining the water takes quite a while, as the tank has a very small drain hole. The aquarium holds 50 litres of water and drains at a rate of 3

4 of a litre per minute.

a. Write the rule that represents the amount of water (A) left in the tank after x minutes.

b. Illustrate this situation on a graph.

c. Chad left and came back when there were 20 litres remaining. Use your graph to determine how long Chad was gone for.



3. The Whitlock Golf and Country Club and The Summerlea Golf Club are private golf courses that are in close proximity to each other. As with many private golf courses, they are struggling financially and are competing with each other for new members. They are offering the following pricing models to attract new members.

Whitlock Summerlea

Membership fee: $1000 Membership fee: $400

Price per game played: $50 Price per game played: $80

What equation can be used to represent the total amount paid (y) by a member at each course given that x represents the # of games played.

Whitlock:

Summerlea:

Wayne is trying to figure out which course would be a better deal for him to join. Please graph both of these equations on the same coordinate plane and use your graph to analyze the situation and answer Wayne’s questions.

a. At what point would the prices at both courses be equal?

b. Under what circumstances is it cheaper to join Whitlock?

Skill Builder



6 Graphing Linear Functions Level 2 EVALUATED

1 4 7y x= − +

My Calculations

2 2

65

y x= −

My Calculations

3 3

52

y x= − +

My Calculations

4 0.2 8y x= −

My Calculations

Graph the following linear functions using a table of values.

Skill Builder


Skill Builder: Graphing Linear Functions Level 2 119

5 0.25 6y x= − +

My Calculations

6 2 3 12 0x y− − =

My Calculations

7 5 4 20x y+ =

My Calculations

8 4 2 0x y+ − =

My Calculations

9 8 5x y− =

My Calculations

10 2 3 8x y+ =

My Calculations




Graphing Linear Functions Level 2

1. Graph each of the following linear functions by creating a table of values and determining the coordinates of two points.

3 4 12x y+ =

x y

1

53

y x= −

x y



2. All that is needed to accurately graph the equation of a line are the coordinates of two points. You have learned how to create a table of values in order to find coordinates to use while graphing. However, given rules that contain fractions or decimals, it is not always easy to find “whole number” points and without integer value coordinates; it is difficult to accurately graph a line.

Develop a strategy that can be used to find integer coordinates using the rule 3 14 2

y x= +

x y



3. Jasmine and Alyssa have each joined separate gyms so that they can have a workout on their ways home from school. The two gyms have different payment plans. During the first month, Jasmine only went to the gym once and paid $52 for the month. In the second month she worked out twelve times and paid $74. Alyssa went to her gym four times the first month, which cost her $54 and ten times the second month and paid $72.

Graph the lines that represent the amounts that Jasmine and Alyssa pay at their respective gyms per month given their number of visits and then use your graph to answer the questions that follow.

a. How much does Jasmine pay per month if she doesn’t work out at all?

b. After how many visits do they each pay the same amount per month?

c. When is it a better deal to attend Jasmine’s gym?

d. Describe each of their payment plans in words

CHAPTER Linear Functions 2 · 2019-02-07 · 2 CHAPTER Skill Builder: Introduction to Linear...

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Transcript of CHAPTER Linear Functions 2 · 2019-02-07 · 2 CHAPTER Skill Builder: Introduction to Linear...