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Flip to reveal answersWhat are the two key features that make a situation linear?
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All 16 Flashcards — Linear models
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Question
What are the two key features that make a situation linear?
Answer
1. Constant rate of change — each unit increase in x produces the same change in y. 2. The graph is a straight line.
Question
When is a linear model the right choice?
Answer
When the data shows a constant rate of change — equal steps in x produce equal steps in y. A scatter plot that looks like a straight line suggests a linear model.
Question
C = 5n + 200 is a cost model. What does each part tell you?
Answer
5n: cost increases by 5 per unit produced (variable cost, the gradient). 200: fixed cost regardless of production level (the y-intercept).
Question
A car travels at a constant speed of 80 km/h. Is distance vs time a linear model? Why?
Answer
Yes — constant speed means equal distance in equal time intervals. Distance = 80t is linear with gradient 80.
Question
You have two data points. How do you build a linear model?
Answer
1. Calculate gradient: m = (y₂ − y₁)/(x₂ − x₁). 2. Use y = mx + c with one point to find c. 3. Write the model.
Question
A model gives T = −2.5t + 80. Find T when t = 12.
Answer
T = −2.5(12) + 80 = −30 + 80 = 50.
Question
Temperature falls from 60°C to 20°C over 8 hours. Write a linear model for T in terms of t.
Answer
m = (20 − 60)/8 = −5. Using (0, 60): T = −5t + 60.
Question
IB asks "Write a linear model." What must your answer include?
Answer
The full equation in y = mx + c form, with numerical values for m and c, using the variables named in the context.
Question
P = 4.5t + 120 (P = population, t = years). Interpret the gradient 4.5.
Answer
The population increases by 4.5 people per year.
Question
W = 0.3d + 50 (weight W kg, distance d km). Interpret the y-intercept 50.
Answer
The initial weight is 50 kg — the weight at the start (d = 0), before any distance has been covered.
Question
IB asks "Interpret the gradient in context." How do you get full marks?
Answer
State: the numerical value, the units, and what it means for the specific context. E.g. "The water level rises by 3 cm per hour."
Question
A linear model has a negative gradient. What does this tell you?
Answer
The quantity is decreasing at a constant rate as the input variable increases.
Question
What does it mean for a linear model to be "valid"?
Answer
The model gives reliable, meaningful predictions for x-values within the range of the original data (interpolation). Outside this range, the model may break down.
Question
IB asks "Is the model valid for x = 50? Give a reason." How do you answer?
Answer
Check if x = 50 is within the data range. If yes: "Yes — x = 50 is within the data range so the estimate is reliable (interpolation)." If no: "Less reliable — x = 50 is outside the data range (extrapolation)."
Question
T = −2t + 100 predicts T = −100 at t = 100. Why is this problematic?
Answer
Physically extreme or impossible values signal model breakdown — this is extrapolation far beyond the data range. Real temperatures may not follow this pattern at t = 100.
Question
What is the key difference between interpolation and extrapolation?
Answer
Interpolation: predicting within the data range — generally reliable. Extrapolation: predicting outside the range — less reliable, the pattern may not continue.
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