Practice Test 7 - AP Precalculus

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

Using the polynomial provided, a quadratic $s(x)=x^2+2x+10$ has $\Delta=4-40=-36$ ; calculate the complex zeros via the quadratic formula and simplify $\sqrt{-36}=6i$ to exact $a\pm bi$ form.
In beam stress analysis, $S(x)=\tfrac12x^4-3x^2+\tfrac{5}{2}$ ; what does its end behavior suggest as $x\to\pm\infty$ ?
A particle's position is $x(t)=\ln(t+1)$ and $y(t)=t^2$ , where $t$ is in seconds and $t\ge 0$ . The slope is $dy/dx=(dy/dt)/(dx/dt)$ . What is the rate of change of $y$ with respect to $x$ at $t=1$ ?
Consider $f(x)=\dfrac{3x^2+1}{x^2-4}$ . Assume $x\neq\pm2$ . Which of the following best describes the limit of $f(x)$ as $x$ approaches infinity (and thus the horizontal asymptote)? Use leading-term reasoning to evaluate $\lim_{x\to\infty} f(x)$ .
The function $f(x) = \sin(x)$ can also be expressed as a phase-shifted cosine function. Which of the following expressions is equivalent to $f(x)$ ?
A wildlife reserve begins with 3,200 deer and grows 4% each year, modeled by $N(t)=3200\cdot(1.04)^t$ . How long will it take for the population to reach 4,000 deer?
### Semi-Log Scales In a semi-log graph, one axis is logarithmic and the other is linear. When the $y$ -axis is logarithmic (base 10), moving up one major tick might represent multiplying $y$ by 10. ### Why Use Semi-Log Plots? Semi-log plots are especially helpful for exponential relationships. - They can turn a curved exponential trend into a straight line. - This makes it easier to estimate whether the percent growth rate stays constant. ### Scenario: Population Growth A city plots $N$ versus $t$ on a semi-log graph (logarithmic $y$ -axis). The points form a straight line, suggesting $N(t)=N_0(1+r)^t$ is reasonable. ### Interpreting a Straight Line A straight line on this plot means equal time steps correspond to equal multiplicative changes in $N$ . In the example of population growth, which statement best describes the trend shown by the semi-log plot?
A circle is centered at the origin and has a radius of 4. The terminal ray of an angle $\theta$ intersects the circle at a point P in Quadrant III with a y-coordinate of $-2$ . What is the value of $\cos\theta$ ?
A rational function is a quotient of polynomials. For $f(x)=\dfrac{x^2-4}{x-2}$ , zeros occur where $f(x)=0$ and the function is defined; vertical asymptotes come from non-canceling denominator zeros. Since $x^2-4=(x-2)(x+2)$ , the function behaves like $x+2$ except at $x=2$ , where it is undefined. Based on the function described, which statement best describes the behavior at its zero?
An investment account grows quickly at first, but the bank adds smaller bonuses after the balance gets large. Model 1: Exponential, $y=1000(1.08)^t$ (base $1.08$ ). Model 2: Logarithmic, $y=1000+900\log(0.5t+1)$ (coefficient $900$ ). The first model matches constant-percent interest, while the second matches diminishing returns from capped bonuses. Based on the models described in the passage, which model best predicts long-term behavior in the context described?
Which of the following represents the annual decay factor for this isotope?
A polynomial function of degree 4 is known to have a global minimum. Which of the following must be true?
In an economic model, price and demand are parameterized by time: $p(t)=50+2t$ (dollars) and $D(t)=200-5t^2$ (units), with $t$ in weeks. The rate of change of demand is $dD/dt$ . What is the rate of change of demand at $t=3$ ?
In environmental modeling, two species populations $x$ and $y$ satisfy $x+y+\theta xy=5$ , where $\theta$ is a constant interaction parameter. As conditions change, $y$ varies with $x$ along this curve. Based on the scenario, differentiate the implicit function with respect to $x$ and find an expression for $\dfrac{dy}{dx}$ .
A car's velocity is modeled by the polynomial $p(t)= -t^3+6t^2$ (m/s) for $0\le t\le 6$ . Rate of change means the average change in velocity per second over an interval. From $t=0$ to $t=2$ , the car speeds up as $p(t)$ rises. From $t=2$ to $t=4$ , the increase slows because the added velocity each second shrinks. From $t=4$ to $t=6$ , velocity decreases, indicating deceleration. Engineers also track sensor error with $r(t)=\frac{12}{t-3}+8$ , which has a vertical asymptote at $t=3$ . Near $t=3$ , small time changes cause large jumps in $r(t)$ . Far from $t=3$ , $r(t)$ levels toward 8, so its rate of change diminishes. Based on the passage, what is the average rate of change of $p(t)$ on $[4,6]$ ?
In an economics model, equilibrium $(q,p)$ satisfies $F(q,p,m)=q^2+p^2-m^2=0$ , where $m>0$ is an income parameter. As $m$ changes, $p$ changes with $q$ held constant. Using the information provided, how does the parameter $m$ affect the derivative $\dfrac{dp}{dm}$ (with $q$ constant)? Assume $p\neq 0$ .
An investment grows from $5000$ to $8000$ at $6\%$ interest compounded quarterly. The model is $8000=5000\left(1+\frac{0.06}{4}\right)^{4t}$ . Apply the change of base formula to express $t$ using natural logarithms. Using the given information, which expression equals $t$ ?
A line segment from (0,0) to (4,6) is traced by the parametrization $x(t)=4t, y(t)=6t$ for $0 \le t \le 1$ . Which parametrization traces the same segment but takes 2 seconds to complete?
Based on the logarithmic expression provided, simplify $\ln\!\left(\tfrac{e^3\sqrt{x}}{x^2}\right)$ for $x>0$ .
A rational function is a quotient of polynomials. For $f(x)=\frac{x^2-4}{x-2}$ , zeros are $x$ -values where $f(x)=0$ , and vertical asymptotes occur where the denominator is zero after simplification. Since $x^2-4=(x-2)(x+2)$ , the function simplifies to $f(x)=x+2$ for $x\ne2$ . The canceled denominator factor means $x=2$ does not create a vertical asymptote; instead, the function has a removable hole there. Using the function provided, what are the zeros of $f(x)=\frac{x^2-4}{x-2}$ ?

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