AM2 - Bookletmisshamillsclasses.weebly.com/.../am2_-_booklet_.pdf · 4!General(Mathematics((Preliminary(Course)(|(Modelling(Linear(Relationships((AM2)(To(find(the(equation(of(aline(from(agraph(or(table(of(values:

1 General Mathematics (Preliminary Course) | Modelling Linear Relationships (AM2)

Algebraic Modelling (AM2)

Modelling Linear Relationships

Name .....................................................................................................

G. Georgiou


• Calculate the Gradient of a Straight Line from a Graph • Determine the Y-‐Intercept for a Given Graph The gradient of a linear graph is given by the coefficient of x. It refers to the slope of a

straight line and is commonly referred to as runrise .

The y-‐intercept of a linear graph is given by the constant (or the number on its own). The y-‐intercept is the point where the line cuts the y-‐axis. Determine the gradient and the y-‐intercept for the following equations.

Coefficient / Gradient Constant / Y-‐Intercept

y = 2x + 3

y = 3 – 3x

y = 7 – x

y = 3x + 4

y = – x+ 3

y = 4 – 3x

Formula

Example 1

A linear graph can be represented by the equation:

y = mx + b

where m is the gradient and b is the y-‐intercept

m = !"#$%&'()&*'+,")%+)-./%$%.+*.#%0.+$'()&*'+,")%+)-./%$%.+

PROVIDED ON HSC FORMULA SHEET

This formula is NOT provided on the HSC formula sheet.


Coefficient / Gradient Constant / Y-‐Intercept

y = –2x + 2

y = x – 3

y = –5x – 1

y = 1 – 2x

y = 2x – 1

y = !"! +"

y = !!!"

To determine the equation of a straight line, you need 2 key pieces of information.

1. Gradient – find two points and calculate the !"#$%&'()&*'+,")%+)-./%$%.+*.#%0.+$'()&*'+,")%+)-./%$%.+

.

2. Y-‐Intercept – find where the graph cut the y-‐axis.

Write the equation for the following straight lines.

Gradient Y-‐Intercept Equation (y = mx + b)

3 2

– 4 – 3

– 1 0

!" – 5

!" 0.25

Example 2


To find the equation of a line from a graph or table of values:

1. Calculate the gradient. a. Choose 2 coordinates of the line and label them !!" #""$ and !!" #""$ . b. Use this formula:

2. Calculate the y-‐intercept. ~ In a graph, find the y value of the coordinate where the line cuts the y-‐axis. ~ In a table of values, find the y value of the coordinate when x = 0. If these 2 options are not available:

a. Form the equation y = mx + b and replace m with your answer to step 1. b. Replace x and y with any coordinate of the line. c. Solve for b.

3. Write your final equation with the values for m and b found in step 1 and 2.

Determine the equation of the following graphs. (a) (b)

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Example 3

Formula

! ="! ! ""#! ! #"

NOT ON FORMULA SHEET


.................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. (c) (d)

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………………………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………………………………. Determine the equation relating variables x and y. (a) (b) .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... .................................................................

Example 5

H.S.C. Question (4)


.................................................................... ................................................................. .................................................................... ................................................................. (c) (d) .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... ................................................................. .................................................................... .................................................................


• Generate Tables of Values from a Linear Equation • Graph Linear Functions with Pencil and Paper, and with Technology, given an

Equation or a Table of Values • Sketch Graphs of Linear Functions Expressed in the Form y = mx + b without the Use

of Tables To graph a linear function:

1. Complete a table of values using appropriate values for x. 2. Draw a suitable number plane based on the values in your table of values. 2. Use each column of the table as a pair of coordinates and plot those points on the number plane. 3. Draw a straight line through the points.

Note: Don’t forget to use a pencil and ruler, draw arrows on all lines, label everything and use a good scale (RALPS) Sketch the lines of the following equations. (a) (b)

Example 6


(c) Sketch y = 2x + 1.

Example 7

Activity Ex 7.01 ALL


• Identify Independent and Dependent Variables in Practical Contexts • Establish a Meaning for the Intercept on the Vertical Axis in a Given Context • Use Linear Equations to Model Practical Situations (Eg. Simple Interest) • Describe the Limitations of Linear Models in Practical Contexts So far, we have only worked on linear functions in mathematical contexts. Now we will apply linear functions to ‘real world’ scenarios.

~ The gradient represents the rate of change between two variables ~ The y-‐intercept represents the initial value of a function When we deal with linear functions, we always have 2 types of variables. An independent variable is ......................................................................................................... .................................................................................................................................................... A dependent variable is .............................................................................................................. .................................................................................................................................................... Generally speaking, the independent variable always goes on the horizontal (x) axis and the dependent variable goes on the vertical (y) axis. The graph below represents the cost of a wedding for a varying number of guests. (a) Calculate the value of the gradient. .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 8

Independent variable is on the x-‐axis

(you choose how many people you invite to

your wedding)

Dependent variable is on the y-‐axis

(the cost of a wedding depends on how many

guests you have)


(b) Explain what the gradient represents in relation to the two variables. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (c) What is the vertical intercept (y-‐intercept) of this line? ...................................................... (d) Explain what the vertical intercept represents in the context of this question. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (e) Identify the dependent and independent variables. .................................................................................................................................................... .................................................................................................................................................... (f) Hence write down the equation for this linear model. .................................................................................................................................................... (g) Identify ONE limitation of this model. .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................


Stacey operates a printing business. When she quotes a price for printing business cards she charges an initial fee to cover the cost of the design and an additional fee based on the number of cards required. The price she charges for the cards is shown on the graph below.

(a) Determine the value for the vertical intercept and explain what this represents. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (b) What is the gradient of the line? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 9

Number of Business Cards


(c) What does the gradient represent in the context of this question? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (d) Write an equation that relates the price (P) to the number of cards printed (n). .................................................................................................................................................... (e) How much would Stacey charge to print and design 500 cards? .................................................................................................................................................... .................................................................................................................................................... (f) If someone was charged $900, how many cards should have been printed? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................


.................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

H.S.C. Question (10)


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Activity Ex 7.03 Q 2, 4, 5, 8


• Using Graphs to Make Conversions from One Measurement to Another

A conversion graph makes it easy to convert between units.

This graph allows for conversion between oC and oF. (a) Convert 20oC to oF. ...................................................................................... (b) Convert 77oF to oC. ...................................................................................... (c) The temperature of a pot of water increased from 15oC to 30oC. By how many degrees Farenheit did the temperature increase? .........................................................................................

.................................................................................................................................................... .................................................................................................................................................... (d) What is the freezing temperature of water in degrees Fahrenheit? .................................................................................................................................................... .................................................................................................................................................... (e) Calculate the rate of change between the two variables. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 13


What is 300 Australian dollars in British pounds? .................................................................................................................................................... How many Brunei dollars are equal in value to 50 British pounds? .................................................................................................................................................... ....................................................................................................................................................

Activity Ex 7.05 Q 2, 3, 4, 6, 7

Example 14

Example 15


• Sketch the Graphs of a Pair of Linear Equations to Find the Point of Intersection • Find the Solution of a Pair of Simultaneous Linear Equations from a Given Graph • Solve Practical Problems using Graphs of Simultaneous Linear Equations When we draw two linear equations on the same axes they most likely will intersect (unless the two equations are parallel to each other). The x and y coordinate where the two lines meet is the solution to the pair of simultaneous equations. In the General Mathematics course, we only solve simultaneous equations by graphing. Write down the solution of the simultaneous equations

P = 8 – 2n and P = n + 2. ........................................................................................ ........................................................................................ (a) Complete the two tables of values for y = 2x -‐ 1 and y = 5 -‐ x.

(b) Sketch the graphs for y = 2x – 1 and y = 5 – x.

(c) Hence write down the solution to the pair of simultaneous equations. .....................................................

Example 16

Example 17


On the Easter long weekend, Dane and Mitchell both left town and headed north to go camping. When Dane left town he travelled at 40 km / hr. Mitchell left an hour after Dane and took a different route. He was able to travel at 60 km / hr. The graph shows the distance they had travelled t hours after Dane left town.

(a) Explain why D = 40t represents the distance Dane has travelled from the town after t hours. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (b) What is the simultaneous solution of the equations D = 4t and D = 60 (t – 1), and what does it represent in this context? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 18


The Year 12 Fund Raising Committee at GSCC is raising money for their school formal. To do this they decided to make memento key rings and sell them. A manufacturer has quoted to the committee an initial charge of $90 plus an additional $4 per key ring. The committee is planning on selling the key rings at $10 each. (a) Write an expression for the total cost C to have n key rings manufactured. ................................................................................................................................................... (b) Write an expression that represents the revenue the committee will receive from selling n key rings. ................................................................................................................................................... (c) Complete these two tables of value showing the cost and revenue from buying and selling n key rings.

(d) Graph the above two expressions on the grid provided below.

Example 19


(e) How many key rings will the committee need to sell to break even? ................................................................................................................................................... ................................................................................................................................................... (f) How much profit would the committee make if it sells 50 key rings? ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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Activity Ex 7.06 Q 1, 2, 3, 4, 7



• Using Stepwise and Piecewise Linear Functions to Model Situations Encountered in Daily Life

Stepwise Graphs Consider the following parking costs at a local shopping centre. No single straight line could be used to draw this graph.

Time Spent Cost 0 -‐ 3 hours Free > 3 -‐ 4 hours $5.00 > 4 -‐ 5 hours $8.00 > 5 -‐ 6 hours $12.00 Over 6 hours $15.00

(a) How much does it cost to park at the airport for 80 minutes? ........................................... (b) How do you know if $6 or $10 is being charged for exactly 30 minutes parking? ................................................................................................................................................... ................................................................................................................................................... (c) When Sharon left the car park she paid $14 parking. How long could have Sharon been in the car park? .................................................................................................................................................... (d) Sharon left the car park at 2:15pm and paid $16 in parking. What is the earliest time she could have arrived at the car park? ....................................................................................................................................................

Example 21


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Piecewise Graphs Consider a real estate agent who charges as follows:

o $2000 for any house sold less than $200,000 o $2000 plus 2% for every dollar over $200,000

We would not be able to have ONE straight line to represent this graph as we have varying rates of change (more than one gradient). This graph shows the commission a real estate agent charges for selling properties of different values.

(a) How much commission does the agent charge for selling a property worth $600,000? .................................................................................................................................................... (b) When the agent sold a house the commission was $5000. What was the selling price of the house? .................................................................................................................................................... (c) The slope of the straight line changes at $400,000. Explain why this is the case? .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 23


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Activity Ex 7.07 Q 1, 2, 4, 5


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AM2 - Bookletmisshamillsclasses.weebly.com/.../am2_-_booklet_.pdf · 4!General(Mathematics((Preliminary(Course)(|(Modelling(Linear(Relationships((AM2)(To(find(the(equation(of(aline(from(agraph(or(table(of(values: