Advanced Math · Math
Nonlinear functions
Quadratic and exponential functions: vertex form, zeros, graph transformations, and reading the meaning of a base or a coefficient in a model.
65 questions in the bank · 12 easier · 32 medium · 21 harder · this domain is ≈35% of the section
What to watch for
In vertex form a(x − h)² + k the vertex is (h, k) — h is what is subtracted, so its sign flips. An exponential base is a multiplier, not a percentage: 1.08 means 8% growth.
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.
- The function f is defined by f(x) = (x − 2)² + 9. In the xy-plane, the graph of y = f(x) has its minimum at which point?
- The function f is defined by f(x) = −2(x − 3)² + 6. In the xy-plane, the graph of y = f(x) has its maximum at which point?
- The function f is defined by f(x) = −2(x − 7)² − 8. In the xy-plane, the graph of y = f(x) has its maximum at which point?
- The function f is defined by f(x) = −2(x − 3)² + 4. In the xy-plane, the graph of y = f(x) has its maximum at which point?
Worked examples
easyThe function f is defined by f(x) = (x − 2)² + 9. In the xy-plane, the graph o…
The function f is defined by f(x) = (x − 2)² + 9. In the xy-plane, the graph of y = f(x) has its minimum at which point?
- A. (−2, 9)
- B. (2, 9)
- C. (2, −9)
- D. (9, 2)
Answer: B. (2, 9)
Explanation
Vertex form is a(x − h)² + k with vertex (h, k). Here the squared term is (x − 2), so h = 2 (note the sign flip inside the parentheses), and k = 9. Because a = 1 is positive, the parabola opens up (minimum).
mediumThe function f is defined by f(x) = −2(x + 3)² − 8. In the xy-plane, the graph…
The function f is defined by f(x) = −2(x + 3)² − 8. In the xy-plane, the graph of y = f(x) has its maximum at which point?
- A. (3, −8)
- B. (−8, −3)
- C. (−3, −8)
- D. (−3, 8)
Answer: C. (−3, −8)
Explanation
Vertex form is a(x − h)² + k with vertex (h, k). Here the squared term is (x + 3), so h = −3 (note the sign flip inside the parentheses), and k = −8. Because a = −2 is negative, the parabola opens down (maximum).
hardTwo savings plans both start at $100. Plan A grows by $80 per year. Plan B gro…
Two savings plans both start at $100. Plan A grows by $80 per year. Plan B grows by 30% per year. What is the least whole number of years after which Plan B is worth more than Plan A?
Answer: 8
Explanation
Plan A after t years: 100 + 80t (linear). Plan B: 100(1.3)^t (exponential). Testing whole years, Plan B first pulls ahead at t = 8 ($815.73 vs $740). Exponential growth always overtakes linear eventually — the question is when.