Modeling skills
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Name the five model types in IB AI SL and their general forms.
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2.6.116 cards
Name the five model types in IB AI SL and their general forms.
Linear: y = mx + c. Quadratic: y = ax² + bx + c. Exponential: y = a · bˣ. Power: y = axⁿ. Sinusoidal: y = a sin(bx + c) + d.
Which model type is best for a quantity that grows proportionally to itself (e.g. bacteria doubling)?
Exponential — constant percentage growth = constant ratio between successive values = exponential model.
Which model type produces a repeating (periodic) graph?
Sinusoidal (trigonometric). Tides, temperature cycles, sound waves — any periodic real-world quantity.
A scatter plot shows a clear straight-line pattern. Which model should you choose?
Linear. A straight-line scatter plot is the defining sign of a linear model.
Scatter plot curves upward and passes near the origin. Which two models should you consider?
Power (y = axⁿ) or exponential (y = a · bˣ). The near-origin hint favours power. Compare R² after fitting both.
Scatter plot rises symmetrically then falls, forming a single peak. Which model fits?
Quadratic — single turning point, symmetric parabola shape.
Scatter plot oscillates up and down repeatedly at regular intervals. Which model fits?
Sinusoidal — regular repeating pattern = periodic = trigonometric model.
IB says "Suggest a suitable model and give a reason." How do you get full marks?
Name the model type AND give one clear reason based on the shape or context. E.g. "Exponential, because the data shows a constant ratio between successive values."
Both power and exponential curves go upward. How do you tell them apart?
Power (y = axⁿ): may pass through origin, no horizontal asymptote to the right. Exponential (y = a · bˣ): never passes through origin, has horizontal asymptote y = 0 as x → −∞.
Data: (1, 3), (2, 12), (3, 48). Check if the ratio between successive y-values is constant.
12/3 = 4 and 48/12 = 4. Constant ratio → exponential model.
Power regression R² = 0.91; exponential regression R² = 0.98. Which do you choose?
Exponential — higher R² means it explains more of the variation. Choose the model with the higher R².
IB asks "Explain why exponential is more appropriate than linear." How do you answer?
State that the data shows a constant multiplicative (percentage) growth rate, not a constant additive change — which matches exponential, not linear.
Population doubles every 5 years. Which model is most appropriate?
Exponential — doubling at a constant time interval means a constant ratio between values, which is the defining feature of exponential models.
A ball follows a single arc up and down. Which model?
Quadratic — the path is a parabola. It has one turning point and is not periodic (doesn't repeat).
Electricity use follows the same pattern every 24 hours. Which model?
Sinusoidal — regular repeating cycle with constant period.
Drag force is proportional to the square of speed. Which model?
Power model: F = av², where n = 2.
2.6.216 cards
What are the steps to perform linear regression on a TI-84 GDC?
1. Enter x data in L1, y data in L2. 2. Stat → Calc → LinReg(ax+b). 3. Note a and b from output. 4. Write the equation y = ax + b.
What does the GDC regression output show you?
The best-fit equation parameters (a, b, etc.) and the correlation coefficient r (or R² for non-linear).
IB asks "use the GDC to find the regression equation." What must you write?
The full equation with all parameters to 3 s.f. E.g. y = 2.35x + 4.18. Include what regression type you used if asked.
After running regression, IB says "use your equation to predict y when x = 10." What do you do?
Substitute x = 10 into the regression equation and calculate. Show the substitution clearly.
Data curves upward steeply. Which regression types should you try?
Exponential (ExpReg) and power (PwrReg). Run both and compare R² values.
Data oscillates regularly. Which regression is appropriate?
Sinusoidal regression (SinReg on TI-84).
You run LinReg (R² = 0.61) and ExpReg (R² = 0.95). What should you do?
Use the exponential model — much higher R² means far better fit.
IB gives a data table showing a constant ratio between successive y-values. Which regression?
Exponential regression (ExpReg). Constant ratio is the hallmark of exponential growth/decay.
GDC ExpReg output: a = 2.3456, b = 0.8123 (for y = a · bˣ). How do you write the answer?
y = 2.35 · 0.812ˣ (all values to 3 s.f.).
IB asks "Write down the values of a and b." Do you need to show GDC working?
No — just state the values clearly. "From GDC: a = 2.35, b = 0.812." No algebraic working is needed.
GDC gives LinReg: y = 3.7x − 12.4. Find the predicted y when x = 5.
y = 3.7(5) − 12.4 = 18.5 − 12.4 = 6.1.
Why must regression coefficients be rounded to 3 s.f. in IB answers?
IB expects 3 significant figures unless specified. Using fewer can cause errors in later parts; IB may not award accuracy marks if rounding is too severe.
What does r = 0.99 tell you about a linear regression?
Very strong positive linear correlation. The model fits the data extremely well.
What is the difference between r and R²?
r: Pearson correlation coefficient, ranges from −1 to 1, linear regression only. R²: coefficient of determination, ranges 0 to 1, applies to all regression types. R² = r² for linear.
IB asks "Comment on the reliability of the model." R² = 0.72. What do you write?
The model has a moderate fit (R² = 0.72 — 72% of variation is explained). Predictions may not be highly reliable.
R² = 1 for a regression. What does this mean?
Perfect fit — every data point lies exactly on the regression curve. All predicted values match observed values exactly.
2.6.316 cards
Define interpolation.
Using a model to predict a value for an input that is within the range of the original data. Generally reliable.
Define extrapolation.
Using a model to predict a value for an input that is outside the range of the original data. Less reliable — the pattern may not continue.
Data collected 2010–2020. You predict the value in 2025. Is this interpolation or extrapolation?
Extrapolation — 2025 is beyond the end of the data range.
Which is generally more reliable — interpolation or extrapolation? Why?
Interpolation — we stay within the range where the model was built and validated. Extrapolation assumes the pattern continues, which may not hold in new conditions.
IB asks "Is your estimate reliable? Give a reason." The x-value is within the data range. How do you answer?
"Yes, the estimate is reliable as the value x = [n] is within the data range (interpolation)."
IB asks "Is your estimate reliable?" The x-value is outside the data range. How do you answer?
"The estimate is less reliable as the value x = [n] is outside the data range (extrapolation). The model may not hold beyond the collected data."
A linear model predicts a negative population for t = 100. What does this show?
The model breaks down for large t — populations cannot be negative. The model is only valid within the original data range.
Why might predictions far into the future be unreliable even with a good model?
Conditions change over time (resources, policy, environment). The model was built on past data and assumes the same pattern continues indefinitely.
What is the "valid domain" of a model?
The range of input values for which the model produces meaningful, realistic outputs — usually the range of the original data.
h(t) = −5t² + 20t gives a ball's height. h(5) = −25. Why is this not valid?
Negative height is physically impossible — the ball has already hit the ground. The model is only valid for 0 ≤ t ≤ 4 (while airborne).
How do you check whether a model output is "sensible"?
Ask: Is the output physically possible? Is the input within the data range? Does the result make sense in the context (correct units, realistic magnitude)?
IB asks "State one limitation of this model." What kind of answer is expected?
One reason the model may not be perfectly accurate, e.g. "The model assumes constant growth rate, but this may not hold over long periods as conditions change."
What is the IB-style format for answering "Is this estimate reliable?"
Yes/No + one reason referencing whether the input is within or outside the data range (interpolation vs extrapolation).
Data collected for 0 ≤ t ≤ 10. You predict at t = 8. Write your reliability comment.
"The estimate is reliable as t = 8 is within the data range (interpolation)."
Data collected for 0 ≤ t ≤ 10. You predict at t = 15. Write your reliability comment.
"The estimate is less reliable as t = 15 is outside the data range (extrapolation). The model may not hold beyond the collected data."
IB asks "Suggest one reason why the model may not be appropriate." Give a strong example answer.
"The model assumes exponential growth continues indefinitely, but in reality growth may slow due to limited resources or carrying capacity."
Topic 2.6 study notes
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