The y-values of the curve are given by the function $$y(t) = \cos^2(t)$$. The range of the function $$f(t)=\cos(t)$$ is $$[-1, 1]$$. When we square the values in this range, the outputs are always non-negative. The smallest possible value is $$0^2 = 0$$, which occurs when $$\cos(t)=0$$. The largest possible value is $$(-1)^2 = 1^2 = 1$$. Therefore, the range of $$y(t) = \cos^2(t)$$ is the closed interval $$[0, 1]$$.

For both representations, eliminating the parameter results in the rectangular equation $$y=x^2$$. Thus, they trace the same parabola. For $$C_1$$, as $$t$$ increases, the point moves along the parabola. For $$C_2$$, as $$s$$ increases, the point also moves along the parabola in the same direction. However, the speed of traversal is different. For example, in $$C_1$$, it takes 2 units of time for $$t$$ to go from 0 to 2, covering x-values from 0 to 2. In $$C_2$$, it takes 2 units of time for $$s$$ to go from 0 to 2, covering x-values from $$0^3=0$$ to $$2^3=8$$. Since $$C_2$$ covers a greater distance along the curve in the same parameter interval, it is traversed more quickly.

To find the coordinates of the point at a specific value of the parameter $$t$$, substitute $$t=2$$ into both parametric equations. For the x-coordinate: $$x(2) = (2)^2 - 3(2) = 4 - 6 = -2$$. For the y-coordinate: $$y(2) = 2(2) + 1 = 4 + 1 = 5$$. Therefore, the coordinates of the point on the curve when $$t=2$$ are $$(-2, 5)$$.

The average rate of change of $$y$$ with respect to $$x$$ is given by the formula $$\frac{\Delta y}{\Delta x} = \frac{y(t_2) - y(t_1)}{x(t_2) - x(t_1)}$$. Here, $$t_1=1$$ and $$t_2=4$$. First, find the coordinates at these times. At $$t=1$$, $$x(1) = \sqrt{1} = 1$$ and $$y(1) = 1^2 - 1 = 0$$. At $$t=4$$, $$x(4) = \sqrt{4} = 2$$ and $$y(4) = 4^2 - 1 = 15$$. Now, calculate the average rate of change: $$\frac{\Delta y}{\Delta x} = \frac{15 - 0}{2 - 1} = \frac{15}{1} = 15$$.

The vertical motion is described by the function $$y(t) = t^3 - 3t$$. The direction of vertical motion changes at a point where the function's rate of change is zero, corresponding to a local maximum or minimum. For the polynomial $$y(t)$$, these extrema occur at the critical points. The derivative is $$y'(t) = 3t^2 - 3$$. Setting the derivative to zero gives $$3t^2 - 3 = 0$$, which leads to $$t^2 = 1$$, so $$t = 1$$ or $$t = -1$$. The question asks for the positive value of $$t$$, which is $$t=1$$.

Both parameterizations describe a path on the unit circle because $$x^2+y^2=\cos^2(\theta)+\sin^2(\theta)=1$$. For $$C_1$$, as $$t$$ goes from 0 to $$2\pi$$, the particle makes one full counter-clockwise revolution. For $$C_2$$, as the parameter $$s$$ goes from 0 to $$\pi$$, the angle $$2s$$ goes from 0 to $$2\pi$$. Thus, the second particle also makes one full counter-clockwise revolution. However, the first particle takes $$2\pi$$ units of time to complete the circle, while the second takes only $$\pi$$ units of time. This means the second particle traverses the same path at twice the speed.

This question tests AP Precalculus skills in parametric functions, vectors, and matrices, focusing on circular motion described by parametric equations. Parametric functions use parameters to express coordinates, vectors represent direction and magnitude, and matrices perform transformations. In this scenario, a point on a wheel follows r(t) = ⟨3cos(2t), 3sin(2t)⟩, describing circular motion with radius 3 meters and angular frequency 2 rad/s. Choice A is correct because at t=π/4, we have x = 3cos(2·π/4) = 3cos(π/2) = 0 and y = 3sin(2·π/4) = 3sin(π/2) = 3, giving coordinates ⟨0, 3⟩. Choice D is incorrect because it appears to use t=π/4 directly in the trig functions without the factor of 2, resulting in cos(π/4) = sin(π/4) = √2/2, giving approximately ⟨2.12, 2.12⟩. To help students: Emphasize the role of the coefficient of t in parametric equations as angular frequency, practice evaluating trigonometric functions at key angles, and visualize how the parameter affects position on the circle. Watch for: Forgetting to multiply t by the coefficient inside trig functions, confusion between radians and degrees, and misremembering special angle values.

This question tests AP Precalculus skills in parametric functions, vectors, and matrices, focusing on parametric equations for circular motion with variable angular velocity. Parametric functions use parameters to express coordinates, vectors represent direction and magnitude, and matrices perform transformations. In this scenario, r(t)=⟨3cos(ωt), 3sin(ωt)⟩ describes a rotating arm of fixed length 3, where ω controls the angular velocity. Choice B is correct because increasing ω increases the coefficient of t inside the trigonometric functions, causing the angle to change more rapidly with time, thus increasing the rotation speed. Choice A is incorrect because ω doesn't affect the radius—the coefficient 3 outside the trig functions determines the radius and remains constant. To help students: Emphasize distinguishing between parameters that affect size (amplitude/radius) versus those that affect rate (frequency/angular velocity), practice analyzing how each parameter influences motion, and use animations to visualize parameter effects. Watch for: Confusion between radius and angular velocity parameters, misunderstanding the role of coefficients inside vs outside trig functions, and incorrect geometric interpretations.

This question tests AP Precalculus skills in parametric functions, vectors, and matrices, focusing on projectile motion with quadratic parametric equations. Parametric functions use parameters to express coordinates, vectors represent direction and magnitude, and matrices perform transformations. In this scenario, r(t)=⟨18t, 2+12t-4.9t²⟩ describes projectile motion with constant horizontal velocity 18 m/s and vertical motion under gravity. Choice A is correct because substituting t=1 gives x(1)=18×1=18 and y(1)=2+12×1-4.9×1²=2+12-4.9=14-4.9=9.1, resulting in ⟨18, 9.1⟩. Choice B is incorrect because it miscalculates the y-component as 7.1 instead of 9.1, possibly from an arithmetic error in combining the three terms. To help students: Emphasize careful evaluation of each term in multi-term expressions, practice substituting values systematically, and understand the physical meaning of each term (initial height, initial velocity, gravity). Watch for: Order of operations errors, sign mistakes with the gravity term, and computational errors when combining multiple terms.

This question tests AP Precalculus skills in parametric functions, vectors, and matrices, focusing on understanding parameters in circular motion equations. Parametric functions use parameters to express coordinates, vectors represent direction and magnitude, and matrices perform transformations. In this scenario, a rotating arm follows r(t) = ⟨Rcos(ωt), Rsin(ωt)⟩ where R is the radius and ω is the angular frequency, describing uniform circular motion. Choice C is correct because increasing ω increases the coefficient of t inside the trigonometric functions, causing the angle ωt to grow faster with time, thus increasing the angular speed around the circle. Choice A is incorrect because ω appears inside the trig functions, not as a coefficient of the radius R, so it doesn't affect the circle's size. To help students: Emphasize distinguishing between parameters that affect size (amplitude) versus speed (frequency), practice interpreting coefficients in parametric equations, and use animations to visualize how parameters affect motion. Watch for: Confusing the roles of different parameters, thinking all coefficients affect size, and not recognizing frequency's effect on speed.