Practice Test 6 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

The matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ is associated with which type of linear transformation?
If the polar equation is $r=4\cos(3\theta)$ , how many petals does the graph have?
What is the phase shift of the function $f(x) = -2\sin(4x+\pi)+1$ ?
A savings account earns 4.2% annual interest compounded monthly. An initial deposit of $P_0=2500$ is made, and the balance is modeled by $A(t)=2500\left(1+\frac{0.042}{12}\right)^{12t}$ , where $t$ is in years. A bank statement shows the following balances (rounded): Year 0: $2500$ , Year 1: $2607$ , Year 2: $2719$ , Year 3: $2836$ , Year 4: $2958$ . Using the provided data, what does the initial value in the exponential function represent in this context?
A rotating beacon's intensity model uses the expression $I(\theta)=\sin\theta\cos\theta$ , where $\theta$ is measured in radians. A technician rewrites the model using a single trigonometric function to simplify computations. Use exactly one standard identity and do not approximate. The angle variable remains in radians throughout. Using the identity, simplify the expression for $I(\theta)$ .
Radioactive mass is $N(t)=200e^{-0.4t}$ and $g(x)=\ln(x/200)$ . Based on the scenario, solve for $t$ in $(g\circ N)(t)=-2$ .
Simplification and analysis: $f(x)=\dfrac{x^2-1}{x^2-x}=\dfrac{(x-1)(x+1)}{x(x-1)}$ . Assume the domain is all real $x$ such that $x\neq 0,1$ . Which statement correctly identifies the vertical asymptote and the horizontal asymptote?
What type of graph is represented by the polar equation $r=2+3\cos(\theta)$ ?
A town has $P_0=20{,}000$ residents at $t=0$ years. A linear model adds $600$ residents/year: $P_L(t)=20{,}000+600t$ . An exponential model grows by $3\%$ per year: $P_E(t)=20{,}000(1.03)^t$ . Variables: $t$ in years, $P$ in residents. The linear rate of change is constant, while the exponential rate increases as the population grows. Using the information provided, how does the rate of change differ between the linear and exponential models?
Based on the scenario, a city models GDP (trillions) by $G(t)=0.02t^3+0.3t^2+2t+50$ and investment (trillions) by $I(t)=0.01t^3+0.15t^2+t+10$ , where $t$ is years since 2000. As time passes, both outputs rise, and higher investment supports higher GDP through expanded production capacity. The similar polynomial shapes indicate they accelerate together rather than changing at a constant rate. This helps planners anticipate budget needs as the economy expands. In the context of the example, what happens to GDP if investment increases over time?
A controlled bacterial culture grows quickly in the first hours, but nutrients run low and the hourly increase shrinks. Model 1: Exponential, $y=800(1.3)^t$ (growth factor $1.3$ ). Model 2: Logarithmic, $y=900+500\log(0.7t+1)$ (coefficient $500$ ). The exponential model suits unchecked reproduction, while the logarithmic model suggests growth slows under limits. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?
A town tracks new streaming-service subscribers each month for one year. Model 1: Exponential, $S(t)=120(1.18)^t$ , uses base $1.18$ to represent steady 18% monthly growth. Model 2: Logarithmic, $S(t)=80+140\log(0.6t+1)$ , uses coefficient $140$ to reflect early buzz that slows as the market fills. By month 10, the actual totals rise only a few subscribers per month, not dozens. Based on the models described in the passage, which model best predicts long-term behavior in the context described?
A drone's position is $x(t)=5t+2$ , $y(t)=4t-1$ (meters). Based on these, what is its speed at $t=3$ ?
The expression $\frac{1-\cos(2\alpha)}{\sin(2\alpha)}$ is equivalent to which of the following for all values of $\alpha$ for which the expression is defined?
Consider $p(x)=\dfrac{x^2-x-6}{x-3}$ . What is the location of the hole in $p$ ?
If $t$ is the number of months after the launch, which of the following functions $U(t)$ models the number of users on the platform?
Two different parametric representations of a curve are given by $C_1: x(t) = t, y(t) = t^2$ for $t \in \mathbb{R}$ and $C_2: x(s) = s^3, y(s) = s^6$ for $s \in \mathbb{R}$ . Which of the following statements correctly compares the two representations?
Find $\theta$ if $\sin(\theta)=-\frac{\sqrt{2}}{2}$ and $0\le\theta<2\pi$ .
What is the exact value of $\tan(\arcsin(-\frac{\sqrt{2}}{2}))$ ?
A ball is thrown upward, and its height (in meters) is $h(t)=-4.9t^2+19.6t+1$ for $0\le t\le 4$ , where t is seconds. Based on the scenario, how does the rate of change at the highest point of a projectile differ from the start?

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