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SAT · Math · Advanced Math
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.
Question 1
f(x) = −2(x − 3)² + 50 A network engineer models the number of meters of cable laid per hour, x hours into a shift, with the function above. What is the maximum value of f(x)?
−2
3
32
50
Answer: D. 50
In vertex form a(x − h)² + k with a < 0, the maximum is k = 50, reached at x = 3 (D). A is the leading coefficient, which only sets the width; B is the x-coordinate of the vertex, when the maximum occurs; C is f(0), the y-intercept.
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?
200(1.25)^t
200(0.75)^t
200(0.25)^t
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
What is the remainder when p(x) = x³ + 8x - 1 is divided by x - 2?
Enter your own answer.
Answer: 23
By the Remainder Theorem, the remainder when p(x) is divided by x - (2) is p(2) = 23.
Question 4
A ball is thrown upward from a balcony. Its height above the ground, in metres, t seconds after it is thrown is modelled by h(t) = -5t² + 20t + 1. What is the maximum height of the ball, in metres?
1
2
16
21
Answer: D. 21
The graph of h is a parabola opening downward, so the maximum is at the vertex, where t = -b/(2a) = -20/(2·(-5)) = 2 seconds. The height there is h(2) = -5(4) + 20(2) + 1 = -20 + 40 + 1 = 21 metres. B stops at the time of the maximum rather than the height; C is h(1) = 16; A is the launch height h(0).
Question 5
The graph of y = f(x) passes through the point (4, 5). A guitar-string maker defines g(x) = −f(x + 3) + 1 to model a second pack. Which point must lie on the graph of y = g(x)?
(7, −4)
(1, 6)
(1, −6)
(1, −4)
Answer: D. (1, −4)
For g(x) = −f(x + 3) + 1, use x + 3 = 4, so x = 1; then g(1) = −5 + 1 = −4, giving (1, −4) (D). A shifts left in the wrong direction (x = 7); B omits the reflection (5 + 1); C adds 1 before reflecting, −(5 + 1).
A few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.