03Cmodelling

toc = Linear Modelling =

Mathematical Models
In general, a __**model**__ is a smaller or simpler version of the real thing. (Eg: model car, model plane, model house) The model allows us to see what the real thing looks like, or to understand some features of the real thing being represented.

In a similar way, a **mathematical model** allows us to understand some features of the real situation being represented. It often involves assumptions which make the model simpler than the real situation. A function and its graph can be used as a mathematical model of a real relationship.

Variables
A ** variable ** is a quantity that can vary (or change). It is usually represented by a pronumeral.
 * A ** dependent variable ** depends on some other variable. This means its value is calculated from the other one. It changes because the other variable has changed.
 * An ** independent variable ** changes on its own, or we can change its value directly.

For example, given the function, ** y = 2x + 3 **
 * ** x ** is the __independent__ variable (x can be any real number)
 * ** y ** is the __dependent__ variable (the value of y depends on the value of x)

In a table of values
 * the __independent__ variable is always in the top row
 * the __dependent__ variable is always in the bottom row

In a graph
 * the __independent__ variable is always on the __horizontal__ axis (x-axis)
 * the __dependent__ variable is always on the __vertical__ axis (y-axis)

Linear Modelling
Read the information carefully
 * Identify the two variables: What are the things that are changing?
 * Allocate pronumerals for each of the two variables
 * Identify the __dependent__ variable: Which variable depends on the other one? (equivalent to y)
 * Identify the __independent__ variable: Which variable changes on its own, or we can change its value directly? (equivalent to x)

Now make a rule connecting the two variables in the form ** y = mx + c ** where
 * instead of ** y ** you put the dependent variable
 * instead of ** x ** you put the independent variable

Use the same process we learned to make a rule for a linear graph.
 * 1) Find the gradient (m)
 * 2) Find the y-intercept (c)
 * 3) Write in the form y = mx + c

Finding the gradient
 * If you are given a __**rate**__ (eg price is $10 __per__ km, or speed is 35 km __per__ hour) then that rate __is__ the gradient.
 * If you are given two or more points then select two points and use the rule for gradient:

Finding the y-intercept
 * If you are given an **__initial value__** plus a rate, then the initial value __is__ the y-intercept
 * If you are given two or more points then
 * write the information you know in the form ** y = mx + c **
 * select one of the points and substitute ** x ** and ** y **
 * solve for c

Write the equation
 * Write in the form: ** y = mx + c **
 * Use the pronumerals you selected instead of **x** and **y**

** Example 1 **
A taxi has the following pricing guide on its window

Determine a linear equation connecting the price and the distance travelled.

__**Solution:**__

The two variables are distance (in km) and price (in $)
 * Let D = distance
 * Let P = price

Price is calculated from distance so
 * D is the independent variable (x)
 * P is the dependent variable (y)

__**Finding the rule**__


 * 1: Find the gradient (m) **

math \\ . \qquad \textit{select } (D = 5,\; P = 20) \text{ and } (D = 10,\; P = 30) \\ .\\ . \qquad m = \dfrac{y_2 - y_1}{x_2 - x_1} \\ .\\ . \qquad \quad = \dfrac{30-20}{10-5} \\. \\ . \qquad \quad = \dfrac{10}{5} \\. \\ . \qquad \quad = 2 math


 * 2: Find the y-intercept (c) **

math \\ . \qquad \textit{write the information we know in the form: } y = mx + c \\. \\ . \qquad P = 2D + c \\ math

math \\ . \qquad \textit{select } (5, \; 20) \; \textit{ so substitute } D = 5 \textit{ and } P = 20 \\. \\ . \qquad 20 = 2 \times 5 + c \\. \\ . \qquad 20 = 10 + c \qquad \{-10\} \\. \\ . \qquad c = 10 math


 * 3: Write the rule in the form: y = mx + c **

math . \qquad P = 2D + 10 math

** Example 2 **
A tradesman charges $125 for a visit plus $55 per hour. Determine a rule for the price the tradesman charges based on the length of the visit. Draw a graph showing the relationship between time and price.

__**Solution:**__

The two variables are time (in hours) and price (in $)
 * Let T = time
 * Let P = price

Price is calculated from time so
 * T is the independent variable (x)
 * P is the dependent variable (y)

__**Finding the rule**__

** 1: Find the gradient (m) **

math \\ . \qquad \textit{From the question, } \text{Rate } = 55 \\. \\ . \qquad \textit{So } m = 55 math


 * 2: Find the y-intercept (c) **

math \\ . \qquad \textit{From the question, } \text{Initial Price } = 125 \\. \\ . \qquad \textit{So } c = 125 math


 * 3: Write the rule in the form: y = mx + c **

math . \qquad P = 55T + 125 math

__**Graph:**__
 * T is independent so goes on horizontal axis
 * P is dependent so goes on vertical axis
 * Not interested in negative time so only draw 1st quadrant

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