AP Pre-Cal - Unit 3 Progress Check: MCQ Part A

​The functions f and g are given by f(θ) = cos⁡θ and g⁡(θ) = sin⁡θ. On which of the following intervals are both f and g increasing?

D - 3π/2 < θ < 2π

The table gives six ordered pairs. A sinusoidal regression y = a sin (bx + c) + d is used to model these data with a function f. To the nearest tenth, what is the value of f(6) predicted by the sinusoidal function model using all of these data?

A - -3.9

The table gives characteristics of a trigonometric function f for selected intervals of θ. Which of the following could define f(θ) ?

A - cosineθ only

In the xy-plane, the terminal ray of an angle of measure θ in standard position intersects the unit circle at point P. If the x-coordinate of P is 2 times the y-coordinate of P, and 0≤θ≤π/2, what is the value of θ?

A - 0.464

The depth of water in a bay periodically increases and decreases with a period of 12 hours. The depth of water, in feet, at a point in the bay can be modeled by a sinusoidal function f of time t, measured in hours starting at t = 0. The depth of water at the point is 10 feet at time t = 3 hours and 2 feet at time t = 9 hours. If these time-depth pairs represent a maximum value and the first minimum value after the maximum, respectively, which of the following could define f(t)?

A - 4cos(π/6(t - 3)) + 6

The periodic function f has a period of 4. The function is increasing on the input-value interval 0,4. Which of the following is true?

A - f(9) < f(15), because f(9) = f(1), f(3) = f(15), and f(1) < f(3).

The figure shows the terminal ray of an angle in standard position intersecting the unit circle at point P in the xy-plane. Segment PQ Is perpendicular to the x-axis. A triangle is formed by the points at the origin, P, and Q. The function A gives the area of the triangle, A (θ), based on the measure of the angle , in radians. What is the value of A (3) ? (Note: The area of a triangle with base b and height h is A = 1/2bh.)

A - sqrt(3) / 8

​The figure shows a terminal ray of an angle of measure θ in standard position intersecting the unit circle at point P in the xy-plane. As θ increases from 0 to 2⁢π, which of the following gives the graph of the displacement of P from the x-axis as a function of θ?

A - the graph that first goes up, then down, then back up to the x axis

If the function ℎ has a period of 3, which of the following could be a portion of the graph of ℎ in the xy-plane?

C - the graph where every 3 units it is equal

​The figure shows a circle in the xy-plane with center at the origin, an angle of measure θ in standard position, and the intersection point of the terminal ray (-3, -4). What is the value of sin⁡θ ?

B - -4/5

The functions f and g are given by f⁡(θ) = 2⁢cos⁡ θ and g⁡(θ) = 2⁢cos⁡(bθ), where b is a constant. If the period of g is half the period of f, then solving which of the following equations will give the correct value of b?

B - 2π/b = 1/2(2π)

A pendulum swings back and forth, toward and away from a point on a wall. The motion of the pendulum is periodic. The distance, in inches, from the point on the wall at time t seconds is modeled by f⁡(t) = 15⁢sin⁡(π⁢t) + 18. What is the least interval of time between the moments when the pendulum is farthest from the point on the wall?

B - The least interval of time is the period of the function, which is 2 seconds, because the coefficient of t is π and the period is 2π divided by this coefficient.

​A data set of coordinate pairs is modeled by the function f(θ) = sin⁡θ. If the model f is consistent with the data set, which of the following should describe the dependent variable values on the interval π < θ < 3⁢π/2 ?

B - The values should be decreasing at a increasing rate on the interval.

​In the xy-plane, two angles in standard position have measures of a and b. If π/2 < a < b < π, which of the following is true about tan⁡ a and tan⁡ b?

B - tan a < tan b

​The figure shows an angle of measure θ in standard position in the xy-plane. At point P, the terminal ray of the angle intersects a circle centered at the origin. If angle A measures 2⁢π/3 radians, what is the x-coordinate of P?

C - -3/2

​The graph of the function ℎ is given. Which of the following could define ℎ⁡(x)?

C - 2sin(π(x - 0.5)) + 1

The functions f and g are given by f(θ) = sin⁡θ and g(θ) = sin⁡(θ + π/2). On the interval 0 ≤ θ ≤ 2⁢π, how many solutions exist to f(θ) = g⁡(θ)?

C - Two

​The function ℎ is defined by ℎ⁡(θ) = a cos⁡(bθ) for constants a and b. If ℎ has a period of π and an amplitude of 4, what are the values of a and b?

D - a = 4 and b = 2

The periodic function y=g(x) has a period of 6. It is known that g⁡(1)=g(7)=0, and that g⁡(5)=3 is a relative maximum of g. Which of the following must also be true?

D - g(3) = g(27)

​In the xy-plane, the terminal ray of an angle in standard position intersects the unit circle at point P. Which of the following is true about the cosine of the angle, as the angle measure increases from 0 to π?

​​B - The cosine of the angle decreases, because the horizontal displacement of P from the y-axis decreases over the entire interval.

The function f is given by f(θ) = cos⁡ θ. If all of the other necessary conditions are met, which of the following could be modeled by f?

D - A scenario where the frequency is 1/2π and the initial value for θ = 0 is 1