SAT · Math · Advanced Math

Nonlinear functions

Nonlinear functions is part of Advanced Math in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include interpret an exponential growth model, decay and half-life, vertex of a quadratic in context and quadratic from zeros and a point.

Common traps: reading b as the percent rather than 1 + rate; confusing initial value with growth per period; treating growth as linear.

Sample questions

Question 1
A turbine blade's profile is a parabola that crosses the x-axis at x = −2 and x = 5 and passes through the point (0, −20). Which of the following is an equation of the parabola?
- A.
  y = x² − 3x − 10
- B.
  y = 2x² + 6x − 20
- C.
  y = −2x² + 6x + 20
- D.
  y = 2x² − 6x − 20
Answer: D. y = 2x² − 6x − 20
Write y = a(x + 2)(x − 5); at (0, −20), −10a = −20 gives a = 2, so y = 2x² − 6x − 20 (D). Choice A has the right zeros but a = 1 and misses (0, −20); B has zeros at 2 and −5; C has the right zeros but passes through (0, 20).
Question 2
The amount of medication in a patient's bloodstream is 200 milligrams and decreases by 25% each year. Which expression gives the amount of medication in a patient's bloodstream, in milligrams, t years from now?
- A.
  200(1.25)^t
- B.
  200(0.75)^t
- C.
  200(0.25)^t
- D.
  200(1 - 25t)
Answer: B. 200(0.75)^t
A quantity that decreases by 25% each year is multiplied by 1 - 0.25 = 0.75 each year. After t years the amount is 200(0.75)^t.
Question 3
A paint factory uses two functions to mix colours: f(x) = 3x + 4 and g(x) = x² − 2. What is the value of f(g(2))?
- A.
  10
- B.
  14
- C.
  20
- D.
  98
Answer: A. 10
First g(2) = 4 − 2 = 2, then f(2) = 3(2) + 4 = 10 (A). D computes g(f(2)) = 10² − 2, composing in the wrong order; C multiplies f(2) × g(2) = 10 × 2; B writes f(g(x)) as 3x² + 2, multiplying only x² by 3, and then gets 3(4) + 2.
Question 4
A 480 g sample of a radioactive isotope has a half-life of 5 years. How many grams of the isotope remain after 20 years?
Enter your own answer.
Question 5
A trampoline park prices a session with f(x) = 2x + 3 and g(x) = x² − 4. What is the value of g(f(g(2)))?
Enter your own answer.

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