What types of situations can be modelled using a linear model?

Any situation that has a constant rate of change can be modelled by a linear model. For example, taxi charges, mobile phone bills, car rental fees, or distance travelled at constant speed.

What is a limitation of linear models?

Linear models continuously increase (or decrease ) at the same rate . This may not reflect real-life situations where there could be a maximum or minimum value for what is being modelled.

True or False? The symmetry of a quadratic function can be a limitation of using it to model a real-life situation.

True. The symmetry of a quadratic function can be a limitation of using it to model a real-life situation. Real-life data is often not perfectly symmetrical .

True or False? A cubic model can be appropriate when the graph of the data being modelled has exactly one maximum and one minimum within an interval

True. A cubic model can be appropriate when the graph of the data being modelled has exactly one maximum and one minimum within an interval

What is a limitation of cubic models ?

A limitation of cubic models is that they have no global maximum or minimum . In real life there is often a maximum or minimum possible value. To overcome this limitation, an appropriate domain can be specified for the cubic function being used to model the situation.

What is a limitation of exponential growth models?

A limitation of exponential growth models is that they do not have a maximum , which may not be realistic in real-life situations. E.g. for an exponential function modelling the spread of a disease in a population, there will be a maximum when all members of the population are infected.

What is meant by the term half-life ?

The half-life for a substance is the time taken for the mass of a substance to halve .

What is direct variation ?

Direct variation (or direct proportion) means that as one variable goes up the other goes up by the same factor .

What is inverse variation ?

Inverse variation (or inverse proportion) means as one variable goes up the other goes down by the same factor .

State a limitation of sinusoidal models with regard to the amplitude of the model.

A limitation of sinusoidal models, with regard to the amplitude of the model, is that the amplitude is the same for each cycle . This may not be the case in real-life situations.

True or False? The period of a sinusoidal model always changes over time .

False. The period of a sinusoidal model does not change over time . The period is constant for each cycle . In real life , however, the time to complete a cycle might change over time .

What does the domain represent in mathematical modelling ?

In mathematical modelling , the domain represents the reasonable range of input values , considering the real-life context of the situation being modelled.

How can unknown parameters be found in mathematical models ?

Unknown parameters in mathematical models can be found by forming equations , substituting given values and solving the equations simultaneously. You will frequently use a graphing calculator (GDC) to help with this.

What is extrapolation in mathematical modelling ?

Extrapolation is making predictions outside the range of the data.

True or False? Extrapolation is an accurate method for predicting new values in mathematical modelling.

False. Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.

What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation ?

A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.

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What types of situations can be modelled using a linear model?

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What types of situations can be modelled using a linear model?
Any situation that has a constant rate of change can be modelled by a linear model.
For example, taxi charges, mobile phone bills, car rental fees, or distance travelled at constant speed.
In a linear model, of the form $f (x) = m x + c$ , what does the $c$ represent?
In a linear model, of the form $f (x) = m x + c$ , the $c$ represents the initial value.
This is the value of the function when $x = 0$ .
E.g. consider the function $C (d) = 4 d + 3$ being used to model a taxi's charges, where $d$ is the number of miles travelled and $C (d)$ is the total cost in dollars of the journey.
The initial value before travelling anywhere, $c$ , is 3 dollars.
In a linear model, of the form $f (x) = m x + c$ , what does the $m$ represent?
In a linear model, of the form $f (x) = m x + c$ , the $m$ represents the rate of change of the function.
E.g. consider the function $C (d) = 4 d + 3$ being used to model a taxi's charges, where $d$ is the number of miles travelled and $C (d)$ is the total cost in dollars of the journey.
The rate of change of the function , $m$ , is 4 dollars per mile travelled.
What is a limitation of linear models?
Linear models continuously increase (or decrease) at the same rate.
This may not reflect real-life situations where there could be a maximum or minimum value for what is being modelled.
Consider a linear function, $h (t) = 25 - 2 t$ , being used to model the height of water, $h (t)$ , in a container, over time, $t$ .
What is a possible limitation of the model and how can it be overcome?
This linear function, $h (t) = 25 - 2 t$ , being used to model the height of water, $h (t)$ , in a container, over time, $t$ , is a decreasing function as the coefficient of $t$ is negative.
This implies that the height of the water in the container is continuously decreasing over time. However, the height of the water cannot actually decrease below zero (i.e. $h (t) ≮ 0$ ).
This problem can be overcome by specifying an appropriate domain, e.g. $t \leq 12.5$ .
What types of situations can be modelled using a quadratic model?
If A quadratic function can be used to model a data set if the graph of the data resembles a $\cup$ or $\cap$ shape.
For example, the vertical height, $h (t)$ of a football $t$ seconds after being kicked can be modelled by a quadratic function.
In a quadratic model of the form $f (x) = a x^{2} + b x + c$ , what does the $c$ represent?
In a quadratic model of the form $f (x) = a x^{2} + b x + c$ , the $c$ represents the initial value of the situation, i.e. $f (0) = c$ .
True or False?
In a quadratic model of the form $f (x) = a x^{2} + b x + c$ where the value of $a$ is negative, the data must have a minimum value.
False.
In a quadratic model of the form $f (x) = a x^{2} + b x + c$ where the value of $a$ is negative, the data must have a maximum value.
For a quadratic function of the form $f (x) = a x^{2} + b x + c$ , what impact does the absolute value of $a$ have on the rate of change of the function?
For a quadratic function of the form $f (x) = a x^{2} + b x + c$ , the absolute value of $a$ has an impact on the rate of change of the function:
- If $a$ has a large absolute value, the rate of change varies rapidly.
- If $a$ has a small absolute value, the rate of change varies slowly.
True or False?
The symmetry of a quadratic function can be a limitation of using it to model a real-life situation.
True.
The symmetry of a quadratic function can be a limitation of using it to model a real-life situation.
Real-life data is often not perfectly symmetrical.
True or False?
A cubic model can be appropriate when the graph of the data being modelled has exactly one maximum and one minimum within an interval
True.
A cubic model can be appropriate when the graph of the data being modelled has exactly one maximum and one minimum within an interval
What is a limitation of cubic models?
A limitation of cubic models is that they have no global maximum or minimum.
In real life there is often a maximum or minimum possible value. To overcome this limitation, an appropriate domain can be specified for the cubic function being used to model the situation.
True or False?
For a cubic model of the form $f (x) = a x^{3} + b x^{2} + c x + d$ , the value for $a$ has the greatest impact on the rate of change of the function.
True.
For a cubic model of the form $f (x) = a x^{3} + b x^{2} + c x + d$ , the value for $a$ has the greatest impact on the rate of change of the function.
What types of situations can be modelled by an exponential function?
Exponential functions can be used to model a situation where there is a constant percentage increase or decrease, such as functions generated by geometric sequences.
For example, the value of a car, $V (t)$ , after $t$ years.
What does "initial" often indicate in exponential modelling problems?
The word "initial" in an exponential modelling question indicates the start of the situation being modelled, e.g. $x = 0$ or $t = 0$ (depending on the variable being used).
To find the initial value of an exponential function, set the variable equal to zero (and simplify the equation if necessary).
True or False?
For a situation modelled by an exponential function of the form $f (x) = k a^{- x} + c$ , a value of $a$ that is greater than $0$ but less than $1$ indicates exponential decay.
False.
For a situation modelled by an exponential function of the form $f (x) = k a^{- x} + c$ , a value of $a$ that is greater than $0$ but less than $1$ does not indicate exponential decay. It represents exponential growth.
For a function of that form, a value of $a$ that is greater $1$ does indicate exponential decay.
For an exponential model of the form $f (x) = k a^{x} + c$ , what does the $c$ represent?
For an exponential model of the form $f (x) = k a^{x} + c$ , the $c$ represents the boundary of the model. This is the value that $f (x)$ gets closer and closer to but can never be equal to.
On the graph, there will be a horizontal asymptote at $y = c$ .
For an exponential model of the form $f (x) = k e^{r x} + c$ , what does the $r$ represent?
For an exponential model of the form $f (x) = k e^{r x} + c$ , the parameter $r$ describes the rate of growth or decay.
The bigger the absolute value of $r$ , the faster the function increases/decreases.
What is a limitation of exponential growth models?
A limitation of exponential growth models is that they do not have a maximum, which may not be realistic in real-life situations.
E.g. for an exponential function modelling the spread of a disease in a population, there will be a maximum when all members of the population are infected.
What is meant by the term half-life?
The half-life for a substance is the time taken for the mass of a substance to halve.
Given an exponential model $f (t) = k a^{- t}$ or $f (t) = k e^{- r t}$ , how is the half-life of a substance found?
Given an exponential model $f (t) = k a^{- t}$ or $f (t) = k e^{- r t}$ , the half-life of a substance is the time $t$ such that $f (t) = \frac{k}{2}$ .
What is direct variation?
Direct variation (or direct proportion) means that as one variable goes up the other goes up by the same factor.
True or False?
If $x^{n}$ and $y$ vary directly (where $n$ is a positive integer), then the ratio $x^{n} : y$ will always be the same.
True.
If $x^{n}$ and $y$ vary directly, then $x^{n} : y$ will always be the same.
What is $k$ used to represent when working with direct variation?
When working with direct variation, $k$ is used to represent the constant of proportionality that connects $x^{n}$ (where $n$ is a positive integer) and $y$ .
$y = k x^{n}$ when $x^{n}$ and $y$ vary directly.
True or False?
A graph of two variables that vary directly with one another can intercept the y-axis at any point.
False.
A graph of two variables that vary directly with one another can not intersect the y-axis at any point.
The graph will always intercept the y-axis at $(0, 0)$ .
True or false?
When solving a direct variation problem, one of the first things you should do is find the value of the constant of proportionality $k$ .
True.
When solving a direct variation problem, one of the first things you should do is find the value of the constant of proportionality $k$ .
What is inverse variation?
Inverse variation (or inverse proportion) means as one variable goes up the other goes down by the same factor.
True or False?
If $x^{n}$ varies inversely with $y$ (where $n$ is a positive integer), then the ratio $x^{n} : y$ will always be the same.
False.
If $x^{n}$ varies inversely with $y$ , then $\frac{1}{x^{n}} : y$ will always be the same.
What is $k$ used to represent when working with inverse variation?
When working with inverse variation, $k$ is used to represent the constant of proportionality that connects $x^{n}$ (where $n$ is a positive integer) and $y$ .
$y = \frac{k}{x^{n}}$ when $x^{n}$ and $y$ vary inversely.
True or False?
The graph of two quantities that vary inversely is a straight line.
False.
The graph of two quantities that vary inversely is not a straight line.
What types of situations can be modelled by a sinusoidal model?
Any situation where the quantity being measured oscillates (fluctuates periodically) can be modelled by a sinusoidal model.
For example, the depth of water at a shore, $D (t)$ , at $t$ hours after midnight.
State a limitation of sinusoidal models with regard to the amplitude of the model.
A limitation of sinusoidal models, with regard to the amplitude of the model, is that the amplitude is the same for each cycle.
This may not be the case in real-life situations.
True or False?
The period of a sinusoidal model always changes over time.
False.
The period of a sinusoidal model does not change over time. The period is constant for each cycle.
In real life, however, the time to complete a cycle might change over time.
For a sinusoidal function of the form $f (x) = a \sin (b (x - c)) + d$ , what happens to the graph as the value of $a$ increases?
For a sinusoidal function of the form $f (x) = a \sin (b (x - c)) + d$ , the $a$ represents the amplitude of the function.
As $a$ increases, the range of the values of the function also increases (the maximums get higher and the minimums get lower).
In a sinusoidal model of the form $f (x) = a \sin (b (x - c)) + d$ , what does the $d$ represent?
In a sinusoidal model of the form $f (x) = a \sin (b (x - c)) + d$ , the $d$ represents the principal axis.
This is the line about which the function fluctuates, and it has the equation $y = d$
In a sinusoidal model of the form $f (x) = a \sin (b (x - c)) + d$ , what does the $c$ represent?
In a sinusoidal model of the form $f (x) = a \sin (b (x - c)) + d$ , the $c$ represents the phase shift.
This is a horizontal translation by $c$ units.
True or False?
In a sinusoidal function of the form $f (x) = a \cos (b (x - c)) + d$ , the greater the value of $b$ , the quicker the function repeats a cycle.
True.
In a sinusoidal function of the form $f (x) = a \cos (b (x - c)) + d$ , the greater the value of $b$ , the quicker the function repeats a cycle.
True or False?
In a sinusoidal function of the form $f (x) = a \sin (b (x - c)) + d$ , any horizontal translations should be applied before any horizontal stretches.
False.
In a sinusoidal function of the form $f (x) = a \sin (b (x - c)) + d$ , any horizontal translations should not be applied before any horizontal stretches.
Horizontal stretches with scale factor $\frac{1}{b}$ should be applied before any horizontal translations $(\begin{matrix} c \\ 0 \end{matrix})$ .
What does the domain represent in mathematical modelling?
In mathematical modelling, the domain represents the reasonable range of input values, considering the real-life context of the situation being modelled.
How can unknown parameters be found in mathematical models?
Unknown parameters in mathematical models can be found by forming equations, substituting given values and solving the equations simultaneously.
You will frequently use a graphing calculator (GDC) to help with this.
What is extrapolation in mathematical modelling?
Extrapolation is making predictions outside the range of the data.
True or False?
Extrapolation is an accurate method for predicting new values in mathematical modelling.
False.
Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.
What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation?
A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.

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Modelling with Functions (DP IB Applications & Interpretation (AI))

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1. Number & Algebra

Number Toolkit

Exponentials & Logs

Sequences & Series

Financial Applications

Complex Numbers

Further Complex Numbers

Matrices

Eigenvalues & Eigenvectors

2. Functions

Linear Functions & Graphs

Further Functions & Graphs

Modelling with Functions

Functions Toolkit

Transformations of Graphs

Modelling with Logarithmic, Logistic & Piecewise Functions

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

Trigonometric Identities & Equations

Voronoi Diagrams

Matrix Transformations

Vector Properties

Vector Equations of Lines

Modelling with Vectors

Graph Theory

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Non-linear Regression

Probability

Probability Distributions

Random Variables

Binomial Distribution

Normal Distribution

Combinations of Normal Distributions & Sample Mean Distributions

Poisson Distribution

Hypothesis Testing using the Chi-squared Distribution

Hypothesis Testing for Population Parameters

Transition Matrices & Markov Chains

5. Calculus

Differentiation

Further Differentiation

Integration

Further Integration

Kinematics

Differential Equations

Coupled & Second Order Differential Equations