SAT Math: Two-variable data and models
Two-variable data: scatterplots, lines of best fit and models.
- Problem-Solving and Data Analysis · Two-variable data: models and scatterplots
- Both modules
- 12 practice questions
- Calculator allowed (Desmos)
What the test covers
- Line of best fit, predicted vs actual
- Linear vs exponential model choice
Key ideas
- Predicted value: substitute \(x\) into the line of best fit. Residual = actual \(-\) predicted.
- Constant differences mean a linear model; constant ratios mean an exponential model.
- The slope of a line of best fit is the predicted change in \(y\) per unit increase in \(x\).
Common mistakes
- A residual is actual minus predicted, not the other way round.
- 'Increasing by 25%' means multiplying by 1.25 each step.
Do it in Desmos
Enter the data in a table and fit y1 ~ mx1 + b or y1 ~ ab^x1 to compare models.
Worked example
Worked example
The line of best fit for a scatterplot is \(y=2.4x+15\). One data point is \((10, 42)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(3\)
- \(39\)
- \(-3\)
- \(42\)
Answer: A: \(3\)
Predicted: \(2.4(10)+15=39\). Actual minus predicted: \(42-39=3\).
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
A quantity \(Q\) takes the values 80, 120, 180 and 270 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Exponential, increasing by 40% per unit of time
- Exponential, increasing by 50% per unit of time
- Linear, increasing by 40 per unit of time
- Linear, increasing by 50 per unit of time
Show the answer and solution
Answer: B: Exponential, increasing by 50% per unit of time
The differences \(40, 60, 90\) are not constant, so it is not linear. The ratios \(\tfrac{120}{80}=\tfrac{180}{120}=\tfrac{270}{180}=1.5\) are constant: exponential growth of 50% per unit of time.
Question 2
The line of best fit for a scatterplot is \(y=0.8x+15\). One data point is \((20, 34)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(31\)
- \(3\)
- \(-3\)
- \(34\)
Show the answer and solution
Answer: B: \(3\)
Predicted: \(0.8(20)+15=31\). Actual minus predicted: \(34-31=3\).
Question 3
The line of best fit for a scatterplot is \(y=0.8x+20\). One data point is \((10, 25)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(-3\)
- \(3\)
- \(25\)
- \(28\)
Show the answer and solution
Answer: A: \(-3\)
Predicted: \(0.8(10)+20=28\). Actual minus predicted: \(25-28=-3\).
Question 4
The line of best fit for a scatterplot is \(y=3.2x+5\). One data point is \((10, 41)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(37\)
- \(41\)
- \(4\)
- \(-4\)
Show the answer and solution
Answer: C: \(4\)
Predicted: \(3.2(10)+5=37\). Actual minus predicted: \(41-37=4\).
Question 5
The line of best fit for a scatterplot is \(y=3.2x+10\). One data point is \((10, 39)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(3\)
- \(-3\)
- \(39\)
- \(42\)
Show the answer and solution
Answer: B: \(-3\)
Predicted: \(3.2(10)+10=42\). Actual minus predicted: \(39-42=-3\).
Question 6
The line of best fit for a scatterplot is \(y=2.4x+15\). One data point is \((25, 77)\). How much greater is the actual \(y\)-value of this point than the value predicted by the line of best fit?
- \(75\)
- \(-2\)
- \(77\)
- \(2\)
Show the answer and solution
Answer: D: \(2\)
Predicted: \(2.4(25)+15=75\). Actual minus predicted: \(77-75=2\).
Question 7
A quantity \(Q\) takes the values 64, 80, 100, 125 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Exponential, increasing by 25% per unit of time
- Linear, increasing by 25 per unit of time
- Linear, increasing by 16 per unit of time
- Exponential, increasing by 16% per unit of time
Show the answer and solution
Answer: A: Exponential, increasing by 25% per unit of time
The differences are not constant, so it is not linear. Each value is \(\frac{5}{4}\) times the previous one: exponential, increasing by 25% per unit of time.
Question 8
A quantity \(Q\) takes the values 32, 48, 72, 108 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Linear, increasing by 16 per unit of time
- Exponential, increasing by 50% per unit of time
- Linear, increasing by 50 per unit of time
- Exponential, increasing by 16% per unit of time
Show the answer and solution
Answer: B: Exponential, increasing by 50% per unit of time
The differences are not constant, so it is not linear. Each value is \(\frac{3}{2}\) times the previous one: exponential, increasing by 50% per unit of time.
Question 9
A quantity \(Q\) takes the values 80, 120, 180, 270 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Exponential, increasing by 50% per unit of time
- Exponential, increasing by 40% per unit of time
- Linear, increasing by 40 per unit of time
- Linear, increasing by 50 per unit of time
Show the answer and solution
Answer: A: Exponential, increasing by 50% per unit of time
The differences are not constant, so it is not linear. Each value is \(\frac{3}{2}\) times the previous one: exponential, increasing by 50% per unit of time.
Question 10
A quantity \(Q\) takes the values 125, 100, 80, 64 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Exponential, decreasing by 20% per unit of time
- Linear, decreasing by 25 per unit of time
- Linear, decreasing by 20 per unit of time
- Exponential, decreasing by 25% per unit of time
Show the answer and solution
Answer: A: Exponential, decreasing by 20% per unit of time
The differences are not constant, so it is not linear. Each value is \(\frac{4}{5}\) times the previous one: exponential, decreasing by 20% per unit of time.
Question 11
A quantity \(Q\) takes the values 250, 150, 90, 54 at times \(t=0,1,2,3\). Which best describes the relationship between \(Q\) and \(t\)?
- Linear, decreasing by 100 per unit of time
- Exponential, decreasing by 100% per unit of time
- Linear, decreasing by 40 per unit of time
- Exponential, decreasing by 40% per unit of time
Show the answer and solution
Answer: D: Exponential, decreasing by 40% per unit of time
The differences are not constant, so it is not linear. Each value is \(\frac{3}{5}\) times the previous one: exponential, decreasing by 40% per unit of time.
Keep going
- Previous topic: One-variable data: center and spread
- Next topic: Probability and conditional probability
- All SAT Math skills · Adaptive SAT Math module simulator · Official SAT practice (Bluebook, Khan Academy)
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