Algebra · Math
Linear functions
Linear functions in context: reading a rate and a starting value out of a model, evaluating a function, or building one from a table or two points.
32 questions in the bank · 4 easier · 22 medium · 6 harder · this domain is ≈35% of the section
What to watch for
The coefficient is a rate — an amount per unit — and the constant is the value when the input is zero. Most wrong answers swap those two roles.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- A moving company charges $50 for a truck plus $6 per mover per job. The function f(x) = 6x + 50 gives the total cost in dollars for a job involving x movers hired. Which of the following is the best interpretation of the coefficient 6 in this context?
- An equipment rental store charges $40 plus $25 per day to rent a power washer. The function f(x) = 25x + 40 gives the total cost in dollars for a job involving x rental days. Which of the following is the best interpretation of the constant 40 in this context?
- A pottery studio charges $80 for materials plus $15 per hour of studio time. The function f(x) = 15x + 80 gives the total cost in dollars for a job involving x studio hours. Which of the following is the best interpretation of the coefficient 15 in this context?
- A dog-walking service charges $120 per month plus $8 per walk. The function f(x) = 8x + 120 gives the monthly cost in dollars for a job involving x walks in a month. Which of the following is the best interpretation of the constant 120 in this context?
Worked examples
easyA moving company charges $50 for a truck plus $6 per mover per job. The functi…
A moving company charges $50 for a truck plus $6 per mover per job. The function f(x) = 6x + 50 gives the total cost in dollars for a job involving x movers hired. Which of the following is the best interpretation of the coefficient 6 in this context?
- A. For each additional one of the movers hired, the total increases by $6.
- B. The total for a job with none of the movers hired is $6.
- C. The total never exceeds $110.
- D. For each additional one of the movers hired, the total increases by $50.
Answer: A. For each additional one of the movers hired, the total increases by $6.
Explanation
In f(x) = 6x + 50, the coefficient 6 multiplies x, so it is the rate of change: every additional unit of x adds $6 to the total. The constant 50 is the starting value when x = 0 — the two numbers answer different questions.
mediumA plumber charges a $25 service fee plus $12 per hour. The function f(x) = 12x…
A plumber charges a $25 service fee plus $12 per hour. The function f(x) = 12x + 25 gives the total charge in dollars for a job involving x hours worked. Which of the following is the best interpretation of the coefficient 12 in this context?
- A. The total for a job with none of the hours worked is $12.
- B. For each additional one of the hours worked, the total increases by $25.
- C. For each additional one of the hours worked, the total increases by $12.
- D. The total never exceeds $145.
Answer: C. For each additional one of the hours worked, the total increases by $12.
Explanation
In f(x) = 12x + 25, the coefficient 12 multiplies x, so it is the rate of change: every additional unit of x adds $12 to the total. The constant 25 is the starting value when x = 0 — the two numbers answer different questions.
hardFor the linear function f, f(1) = 1 and f(3) = −5. What is the value of f(5)?
For the linear function f, f(1) = 1 and f(3) = −5. What is the value of f(5)?
- A. −15
- B. −12
- C. −11
- D. −6
Answer: C. −11
Explanation
The slope is (−5 − 1)/(3 — 1) = −3. From f(3) = −5, each unit step in x changes f by −3, and 5 is 2 steps past 3: f(5) = −5 + 2·−3 = −11.