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Skill Checklist
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
26 Skills Available
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Linear Models and Modeling Skills
3 skills
Mathematical modelling and assumptions
SL AI 2.5
A mathematical model is an equation or graph that represents a real-world situation and can be used to analyze and make predictions about that situation. Mathematical models may be exact or approximate.
Because real-world scenarios usually involve many variables, we often identify the most important ones and making reasonable assumptions about the rest. A good model simplifies the situation as much as possible without significantly reducing the accuracy of its predictions.
In a mathematical model, constants and coefficients are called parameters. The general shape of a model is given by its family (linear, quadratic, exponential, etc.), but the more specific values (like intercepts, asymptotes, or steepness) are controlled by the parameters.
Linear models
SL AI 2.5
A linear model is represented by a straight-line graph.
Since a linear model can be defined by one point and a gradient or two points, they are the simplest models to construct. The most common form of a linear model is y=ax+b, where a is the slope and b is the y-intercept.
Extrapolation
SL AI 2.5
Extrapolation is when we predict values beyond the domain of the given points. Extrapolating may work for certain situations, but it does not work for many others. Pay attention to the context of a model when extrapolating and consider whether the observed behavior is likely to change in the long-run.
Your understanding of extrapolation can be tested by questions that ask you to interpret plausible inputs and outputs.
Example
Between the ages of 5 and 10, the height h cm of children can be modeled by
h=6a+80
where a is their age in years.
Extrapolation would be using this model for a 40 year old, which would give a prediction of h=6⋅40+80=320cm (around 10'6"). This is obviously crazy, and in this case it's obvious why a model for children's height does not apply to adults. But sometimes it's less obvious, and you should always be careful when extrapolating.
Piecewise Linear Models
1 skill
Piecewise Linear Models
SL AI 2.5
We use a piecewise linear model when different linear models apply over different parts of the domain of points. Basically, a piecewise linear model is a collection of smaller, domain-restricted models.
We write piecewise functions with the following notation:
f(x)={g(x),a<x<bh(x),b≤x<c
Quadratic Models
4 skills
Quadratic Models
SL AI 2.5
A quadratic model has a turning point (vertex) at which its minimum or maximum value occurs. The general form of a quadratic equation is ax2+bx+c.
If a<0, the turning point of a quadratic is its maximum; if a>0, the turning point of a quadratic is its minimum.
Fitting a quadratic model
SL AI 2.5
Given 3 pieces of data, we can solve for a, b and c in a quadratic model ax2+bx+c.
Example
The points (1,−25), (−1,−1) and (−3,7) lie on a parabola with equation y=ax2+bx+c. Find a,b and c.
Plugging in the x coordinates and setting equal to the y-coordinates gives 3 equations:
⎩⎪⎨⎪⎧a⋅12+b⋅1+c=−25a⋅(−1)2+b⋅(−1)+c=−1a⋅(−3)2+b⋅(−3)+c=7⇒⎩⎪⎨⎪⎧a+b+c=−25a−b+c=−19a−3b+c=7
Solving this using a calculator gives a=−2,b=−12,c=−11. Thus the parabola has equation
y=−2x2−12x−11
Quadratic x-intercepts
SL AI 2.5
The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.
The equation of the parabola below is −(x−α)(x−β):
Vertex and Axis of Symmetry
SL AI 2.5
The graph of a quadratic function has the general shape of a parabola.
It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.
Cubic Models
1 skill
Cubic Models
SL AI 2.5
Cubic models have the form ax3+bx2+cx+d. Cubic graphs may have 0 or 2 turning points. When cubic graphs have 0 turning points, they have a short flat section where the function appears constant.
Exponential Models
5 skills
Exponential models
SL AI 2.5
An exponential model represents quantities that multiply repetetively by a constant factor b. The basic form of an exponential is bx, but any exponential can be written in the form Abx+k.
The graph of an exponential model is a curve that approaches a horizontal asymptote at y=k on one side, and has a y-intercept at (0,A+k). Because of the asymptote on an exponential graph, exponential models are good at describing behaviors that level off over time.
Exponential growth
SL AI 2.5
Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth is modeled by functions of the form
f(t)=Abt+c,
where b>1. b is called the growth factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.
Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0
Exponential decay
SL AI 2.5
Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay is modeled by functions of the form
f(t)=Abt+c,
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential decay if the instantaneous growth rate, k, is negative.
Concept of half-life
AHL AI 2.9
For any quantity that decays exponentially, the half-life is the amount of time it takes for the quantity to halve in value.
Calculating half-life
AHL AI 2.9
From any exponential decay model of the form f(t)=Abkt (0<b<1), the half-life, or time for the value of f to reach half of its current value, is given by t1/2=−klogb2.
Most commonly, given an equation of the form f(t)=Aekt, the half life is given by −kln2.
Log models, scales and linearization
4 skills
Natural logarithmic models
AHL AI 2.9
A natural logarithmic model is given by f(x)=a+blnx.
Notice f(1)=a and f(e)=a+b.
Logarithmic scales
AHL AI 2.10
When dealing with numbers at very different scales (i.e. 0.000001 and 1,000,000), it can be helpful to express numbers using a logarithmic scale, which converts any number x to logb(x) where b is a common base (often 10).
Example
The chemical pH scale is calculated by
pH=−log([H+])
where [H+] is the concentration of hydrogen ions. The following table give you a sense of how it works:
Linearizing exponential and power models
AHL AI 2.10
Exponential models of the form y=Aekx and power models of the form y=Axn can be linearized by taking logs:
y=Aekx⟺y=Axn⟺lny=lnA+kxlny=lnA+nlnx
Hence, given data in terms of x and y, we can convert the data into lny and lnx.
If lny and x have a linear relationship, then y and x have an exponential relationship.
If lny and lnx have a linear relationship, then y and x have a power relationship.
We can find values for A and k or n by performing a linear regression on lny and x or lnx.
log-log and semi-log graphs
AHL AI 2.10
Both axes of a log-log graph have a logarithmic scale. Straight lines on log-log graphs represent power relationships.
One axis of a semi-log graph has a logarithmic scale. Straight lines on semi-log graphs represent exponential relationships.
Sinuisoidal Models
3 skills
Sinusoidal Models and their features
SL AI 2.5
Sinusoidal models describe quantities that repeat in regular intervals, or periodically, and are of the form y=asin(bx)+c or y=acos(bx)+c.
A sinusoidal curve y=acos(bx)+c is graphed below with key features.
The principal axis, the line around which the sinusoid oscillates, is given by y=c.
The amplitude, or the maximum distance the sinusoid reaches above and below the principal axis, is a.
The period, or the horizontal distance between consecutive maxima, is given by b360° (or b2πrad for HL).
Phase shifts
AHL AI 2.9
If a sinusoidal model has a phase shift, it has been moved horizontally. Now, f(x)=asin(b(x−h))+c, where h is the phase shift.
Sinusoidal regression with technology
AHL AI 4.13
Your calculator should have a function called sinusoidal regression which you can use when you know at least 4 points on a sinusoidal function, and you can estimate the period. To use it, first enter the x coordinates (or independent variables) into L1 and the y coordinates (or dependent variable) into L2.
The calculator will likely ask you to provide a number for "iterations", which is simply the number of "loops" it makes in refining its approximation. 5 will be plenty unless a problem asks for a very high degree of accuracy.
Non-linear piecewise models
1 skill
Non-Linear Piecewise Models
AHL AI 2.9
On the HL test, you may see piecewise models that have non-linear pieces.
For example, f(x)={xx<0x2x≥0is graphed below.
Logistic Models
1 skill
Logistic Models
AHL AI 2.9
A logistic model describes growth that appears exponential for smaller values but slows as it approaches a carrying capacity, represented by a horizontal asymptote above the curve.
Logistic models are given by the general equation f(x)=1+Ce−kxL, where L is the carrying capacity. Logistic models are particularly effective for modelling population growth, as they tend to grow exponentially from small numbers yet have a carrying capacity capped by the scarcity of space, food, and water. k is often called the intrinsic rate, and it represents the rate of growth of a quantity before it nears carrying capacity. Finally, C controls the initial population since f(0)=1+CL, where f(0) is the initial population.
Power Models & Proportionality
3 skills
Direct Proportion
SL AI 2.5
Directly proportional quantities are constant multiples of each other. In the context of modelling, we typically say, "y varies directly with xn," which means y=kxn for some constant k. This can be denoted y∝xn.
If y is directly proportional to xn, then x=0⟺y=0.
If y is directly proportional to xn, then if x increases (or decreases) by a factor of c, y increases (or decreases) by a factor of cn.
Inverse proportion
SL AI 2.5
If y varies inversely with xn, then y=xnk.
If y is inversely proportional to xn (y∝xn1), then the y-axis is an asymptote of the graph of y=f(x).
Fitting a power model
SL AI 2.5
Proportionality relations can be used to build models called power models, which have the form
y=a⋅xb
which is equivalent to saying y∝xb.
Power models can be found from given data using your calculator's power regression feature.