Chapter 5 : Linear Functions

Block A By: Meimenat, Irene, Fion, &

Nicole.

Sections

*Chapter Summaries ( Main Concepts): By: Meimenat

*Chapter Mathematical Skills: By: Irene

*Chapter Questions and Answers By: Fion, Irene, Meimenat and Nicole.

Sections: Summaries: Section 5.1-Slope, Slope Formula, Types of Slope, Zero and Undefined Slope,

X- Intercept & Y-Intercept, Graphing.

Section 5.2- Rate of Change, Identifying Linear Functions, Problem Solving,

Quadrants, Slope , Directions, Distance, Time Graph.

Section 5.3 – X- Intercept, Y- Intercept, One Intercept, Two Intercept, Infinite

Interceps.

Section 5.4- Parallel and Perpendicular Lines.

Section 5.5- Intercepts, Problem Solving, Slope , Domain & Range.

Mathemetical Skills: Section 1. 2. 3. 4. 5.

Questions and Answers: Section 1. 2. 3. 4. 5.

Chapter Summaries: Chapter 5

5.1 Slope ( Main Concepts)

 

What is a Slope?         The Slope of a linear equation describes the steepness and the direction of 

a line; Slope is the rate of change and is represented with the symbol “m”. 

 

There are 4 types of Slope:

 

Figure 1. -Negative Slope.

 

 

Both the X- coordinate and Y- Coordinate are decreasing.  

Figure 2. - Zero Slope  

 

 

Represented by a Horizontal Line. Two different points have the same Y- Value, the line segment joining the two

points is Horizontal.

Figure 3. - Positive Slope

Has both the X- Coordinate and Y-Coordinate increasing.

Figure 4.- Undefined Slope

If two different points have the same X- Values, then the line forming the two points is Vertical.

An Undefined Slope is represented by a Vertical Line. Extra Pointers:

The run between two points on a coordinate plane- represents the change in X- Variables.

The rise between two points on a coordinate system- represents the change in Y- Variables.

5.2 – Rate Of Change ( Main Concepts)

Symbol( s):

1. Δ- Delta- Represents Changes.

Rate Of Changes within Fraction Notation:

2. Km/h- Kilometers/ hour.

3. Mi/Gal- Miles per Gallon.

4. Dollars/hr- Dollars per hour.

Important Pointers:

Δy /Δx = ( change of y / change of x ) – Calculated exactly the same way as

the Slope.

Quadrants:

There are 4 Quadrants on a Coordinate Plane.

I. Quadrant I. - X Value- Positive; Y- Value- Positive. II. Quadrant II. - X Value- Negative; Y- Value- Positive. III. Quadrant III. - X Value- Negative; Y- Value- Negative. IV. Quadrant IV. - X Value- Positive; Y- Value- Negative.

Objectives:

To be able to determine Slopes and the Rate of Change in various situations. Quadrant Identifications. Problem Solving Strategic through the concepts of Rate of Change.

5.3 Graphing Linear Functions ( Main Concepts)

What are needed in order to Graph a Line? 1. Given Coordinates/ Points of a line are required. 2. Given Slope of a line.

It is important to understand the concepts and use the points given within a

problem to estimate the other points on the line given.

Intercepts:

There are two intercepts:

1. Y- Intercept- the point at which a graph crosses the y- axis.

2. X- Intercept: the point at which a graph crosses the x- axis.

Summary:

The X- Intercept of a line is the point ( a,0) where the line intersects the X- Axis. The Y- Intercept of a line is the point ( 0,b) where the line intersects the Y- Axis.

Another way to graph a line is to get it in slope-intercept form:

y = mx + b, whereas,

m is the slope of the line. b is the y-intercept.

The Forms Of Slope Intercepts:

I. One Intercept.

II. Two Intercepts.

III. Infinite Intercepts.

5.4 Parallel and Perpendicular Lines ( Main Concepts)

There are two types of lines:

1. Parallel Lines 2. Perpendicular Lines

I. Parallel Lines Lines within a coordinate system that never intersects. Parallel Lines have Identical Slopes. Rise and Fall at the same rate.

II. Perpendicular Lines

Lines within the coordinate system that forms right angles when they intersect. The product of the slopes of Perpendicular Lines is ( -1). If the Slope of a line is ( a/b ) then the line Perpendicular to it will have a Slope of

( -b/a).

Graphing Perpendicular and Parallel Lines:

I. Parallel Lines

II. Perpendicualr Lines

In Conclusion,

5.4 Applications of Linear Relations

( Main Concepts)

Objective: Focusing on problems involving concepts learned previously such as : Slope –Intercepts, Slopes, Domain and Range, of Linear Relations.

The X- Values could be represented by the following terms:

1. Domain. Definition: The domain of a function is the set of all possible input values (x values), which allows the function formula to work.

2. Input.

- X Values

3. Independent Variable. Definition: a variable in a functional relation whose value determines the value or values of other variables.

The Y- Values could be represented by the following terms:

1. Range. Definition: The range is the set of all possible output values (usually y), which result from using the function formula.

2. Output.

- Y- Values

3. Dependent Variable. Definition: a variable in a functional relation whose value is determined by the values assumed by other variables in the relation.

Key Points:

A change in the independent variable directly causes a change in the dependent variable.

I. Domain & Range

II. Input & Output

III. Independent & Dependent Variables

Chapter 5: The Mathematical Skills needed to be mastered in order to fully understand the concepts listed.

Chapter 5: The Mathematical Skills needed to be mastered in order to fully understand the concepts listed.

Chapter 5 Objectives:

Chapter 5 summarizes the topics of Linear Equations and it involves the Slope, Rate Of Change, Graphing Linear Equations, Parallel and Perpendicular Lines and Applications of Linear Relations.

1. Slope

First, it is essential to understand the different types of slopes by looking at the

steepness and direction of a line. Second, the formula of calculating the slope of a line should be well understood

and memorized. Formula: m= Vertical Change/ Horizontal Change = Rise/ Run.

The skills needed to find the slope of a given line using the formula above requires: ( Note: To understand the change in both ways) m= Y2-Y1 /X2-X1

The skill to find the slope of a line with clear paired points on a coordinate system is needed. ( Note: Especially to match Y2, Y1, X2, X1 with the correct points.)

Third the special lines with specific slope, for example horizontal lines and

vertical lines should be paid attention to. The ability to tell the difference between these two kinds of lines is required. Also it is necessary not to mess up with the lines and their special slopes.

2. Rate Of Change

Recognize the symbol that is used to represent change: Delta( Δ ).

The units in fraction notation should be clearly understood and used in

application questions. ( Km/h ; Mi/ Gal ; Dollars/ Hr ).

The formula to determine the rate of change has to be well understood and

memorized.

Rate of Change: Δy /Δx = Y2-Y1 / X2 – X1

The ability to use this formula using provided data and information is unnecessary.

In visualized graphs, recognize the rate of change quickly . Match Y2, Y1, X2, X1, with the data in the question or on the graph.

3. Graphing Linear Equations

Use the point and slope provided to draw the line on a coordinate system. Know about the intercepts on a coordinate system and tell the difference between

X- Intercepts and Y- Intercepts. X- Intercept ( a, 0) Y- Intercept ( 0, b)

4. Parallel and Perpendicular Lines

Recognize the situations when 2 lines are parallel or perpendicular. Being able to prove that 2 lines are perpendicular or not using the formula

m1=m2 Being able to prove that 2 lines are parallel or not using the formula

m1 x m2 = -1

Being able to think about the formulas between the two parallel or perpendicular lines.

5. Applications of Linear Relations Being able to solve word problems using Linear Relations.

Questions 5.1-5.5 Linear Functions and Slope

Chapter 5 Linear Equations/ Functions

Section1 5.1 (linear functions) 

Question#1: Find the slope of the line containing each pair of points. 

a) (2,3) & (6,9) 

Question#2: Arrange the slope from flattest to steepest. 

a) ‐3/2, 0, 4/3, undefined. 

 

5.2 (Rate of Change) Question#3: A water tower holds 10000 liters of water. When a value is 

opened, the tower has 9700 liters after 30 seconds, and 9400 liters 

after 60 seconds. The water continues to drain at the same rate until 

the tower is empty. 

a) Determine the rate of change of volume with respect to time. 

b) How much water remains after 4 minutes? 

c) How much time is needed to empty ¾ of the water in the tower?  

d) How long does it take to empty the water tower? 

 

5.3 (Linear Functions) Question#4: Which is a greater slope: an incline that is 50%, or one with 

an incline of 100%? 

Question#5: Determine the slope of the line with the given X and Y 

intercept. 

a) (2, 0) (0, 2) 

b) (0, 3) (0, 0)                

Question#6: Determine the slope of the line with the given X and Y 

a) (‐2, 0) (0, ‐2) 

b) (‐4, 0) (0, 0) 

 

5.5 (Parallel and Perpendicular lines)  Question#7: Line A and Line B are parallel, and Line A is 10y=5x+20. If 

line B passes through point (4, 8), what is the linear equation of line B? 

Question#8: Line A and Line B are perpendicular, and Line A is 3y=21x+1. 

If Line B passes through the origin and is parallel to line C which passes 

through (14, 9), what is the linear equation for Line C? 

 

 

5.5 (Applications of Linear Functions)  

 Question#9: Each semester at college, a student must pay ‐‐ costs plus a 

student service fee. To take five courses in a semester costs $3270, to 

take four courses costs $2640. 

a) Find the cost per course. 

b) Find the student service fee. 

c) Find the domain and range. 

Question#10: A new bicycle rental company opened last week. Joe 

rented a bicycle there yesterday to go to the nearby town. When he rode 

16km, he needed to pay $50. When he rode 45km, he needed to pay 

$108. 

a) Find the fee that Joe needed to pay no matter how far he rode. 

b) Find the fee required for driving 1km. 

Answer Key

ANSWER KEY Question#1 

Solution: 

m=3‐9/2‐6  

m=‐6/‐4  

m= 3/2 

Slope = 3/2 

 

 

Question#2 

Solution: 

Flattest→Steepest 

Undefined, 0, ‐3/2, 4/3 

Answer: undefined, 0, ‐3/2, 4/3 

 

 

Question#3 

a) solution: formula: slope=rate of change it run=△time 

Slope =rise/run=△?/△t 

1st of 30s.       Rate=‐300L/30s=‐10 L/s 

Answer: The rare of change of the volume with respect to time is ‐10L/s. 

 

b) solution: After 4min →4×60s=240s 

1min=60s →240s×(‐10Liter/1s)=‐2400L/1000‐2400=7600 

Answer: 7600Liters of water remains after 4min. 

 

c)solution:3/4×10000=7500Leters emptied. 

10Liters/5×time/s=7500Liters 

10t=7500    t=750s÷60=12.5mm 

Answer: 12.5mm is needed to empty ¾ of the mater in the tower. 

 

d) Solution: time=? 

For losing 10000L 

10L/s×t(s) =1000L 

10t=10000 

t=1000(s)÷60=16.7min 

Answer: 16.7min is needed to empty the water tower. 

 

 

Question#4 

Answer: The 50% inclined slope has a larger slope. 

 

 

Question#5 

a) m=2‐0/0‐2=2/‐2=‐1          Answer: m=‐1 

b) m=0‐3/0‐0=‐3/0                Answer: m=undefined 

 

 

Question#6 

a) m=‐2‐0/0‐2=‐2/‐2=1        Answer: m=1 

b) m=2‐0/0+2=2/2=1           Answer: m=1 

 

 

Question#7 

Line A→ y=1/2x+2 

∵Line A∥Line B, 

∴mA=mB 

∴mB=1/2 

And ∵Line B passes through (4, 8) 

∴The linear equation of line B is y=1/2x +6 

 

Question#8 

Line A is y=7x+1/3 

∵Line A⊥Line B and Line B passes through the origin 

∴Line B is y=1/7x 

∵Line B ∥ Line C and Line C passes through (14, 9) 

∴The linear equation of Line C is y=1/7x+7 

 

Question#9 

a) y=mx+b→service fee 

m=rate     x=courses taken (number) 

95, 3270) (4, 2640)=26400‐3270/4‐5=‐630/1 

Answer: $630/course 

b) y=630x+b 

y=2640→2640=63014+b 

2640=2520+b 

b=2640‐2520=120 

c) y=cost of tee 

Domain: x≥0,x integers                         Domain{0,1,2,3,4} 

Range: y≥120                                           Range{120,750,1380} 

 

Question#10 

 

a) X=(108‐50)÷(45‐16)=$2    

b=108‐2×45=$18 

b) X=(108‐50)÷(45‐16)=$2    

 

The End!