SAT Linear functions
Evaluate, build, and interpret linear functions f(x) = mx + b.
How to score it
- Evaluating: substitute and compute; for nested values, work inside-out.
- Building from context: the starting value is b, the per-unit rate is m.
- Watch the “after the first day” offset — it shifts your input by one.
Common traps
- Off-by-one from a hidden “after the first …” offset.
- Swaps slope and intercept when reading the prose.
- Extrapolates past the given points without using the rate.
The 6 question types, with real examples
Evaluate at a given input
“The function f is defined by f(x) = mx + b. What is the value of f(k)?”
The function f is defined by . What is the value of f(3)?
- A✓
- B
- C
- D
Substituting 3 for x in gives f(3) = 5(3) - 8 = 15 - 8 = 7.
Build a linear function from context
“Which function gives the [quantity] after x [units]?”
A tree is 8 feet tall. After the first week, the tree grows at a constant rate of 0.5 feet per week. Which function gives the height h, in feet, of the tree after w weeks?
- A
- B
- C✓
- D
The tree starts at 8 feet. Growth begins after the first week, so after w weeks, the tree has grown for weeks. The function is .
Two-point parametric model
“[Quantity] was Q₁ at P₁ and Q₂ at P₂. Based on the model, what is [quantity] when …?”
The temperature of a chemical reaction is modeled as a linear function of time. The temperature was 18 degrees Celsius at 5 minutes and 42 degrees Celsius at 13 minutes. Based on the model, what is the temperature, in degrees Celsius, at 9 minutes?
- A
- B✓
- C
- D
The slope is (42 - 18)/(13 - 5) = 24/8 = 3 degrees per minute. Using point-slope form with the point (5, 18): . At minutes: (9 - 5) = 18 + 3(4) = 18 + 12 = 30 degrees Celsius.
Interpret slope or intercept of a linear function in context
“[context with linear function f]. What is the best interpretation of [m or b] in this context?”
A truck rental company charges a flat fee of 60 dollars plus 0.85 dollars per mile driven. The function gives the total rental cost, in dollars, for driving m miles. What is the best interpretation of 0.85 in this context?
- AThe cost per mile driven, in dollars✓
- BThe flat rental fee, in dollars
- CThe number of miles that can be driven for 0.85 dollars
- DThe discount per mile driven, in dollars
The function shows that for each mile m driven, an additional 0.85 dollars is added to the cost. The coefficient 0.85 represents the per-mile charge or cost per mile driven.
Find x- or y-intercept from a defined function
“The function [h] is defined by [h(x) = ax + b], with x-intercept (a,0) and y-intercept (0,b). What is the value of [a + b] or [ab]?”
The function g is defined by . What is the value of the sum of the x-intercept and y-intercept of the graph of (x) in the xy-plane?
- A
- B✓
- C
- D
The x-intercept occurs when , so , giving . The y-intercept occurs when , so g(0) = 10. The sum is 5 + 10 = 15.
Constant rate, derived units
“[n] degrees Fahrenheit per [unit]. By how much did the temperature increase, in [other unit]?”
A liquid cooled at a constant rate of 6 degrees Fahrenheit per minute for 25 minutes. By how much did the temperature increase, in degrees Fahrenheit?
- A
- B
- C
- D−150✓
The liquid cooled at 6 degrees Fahrenheit per minute for 25 minutes, so the change is 6 × 25 = 150 degrees. Since it cooled, the increase is −150 degrees Fahrenheit.
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