Practice Test 97 - Algebra 2

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

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1. A circular signal range includes all points within a fixed distance of a transmitter at $(3,4)$. If the range radius is $10$, what is the circle equation for the boundary of the range?
2. Which complex number has modulus $5$? (Use $|a+bi| = \sqrt{a^2+b^2}$.)
3. Use the Rational Zeros Theorem to list all possible rational zeros of $P(x)=3x^3-2x^2+x-6.$ (Recall: any rational zero $p/q$ in lowest terms has $p\mid$ constant term and $q\mid$ leading coefficient.)
4. For the logarithmic function $g(x)=\log_2(x-3)$, find the vertical asymptote and the x-intercept.
5. A company models its daily profit (in dollars) by $P(x)=(x-2)(x+4)$, where $x$ is the number of days since tracking began. On what interval is $P(x)$ negative?
6. A rectangular garden is being designed so that its area (in square meters) is $A(x)=x(30-x)$, where $x$ is the length (in meters) of one side and $30-x$ is the length of the other side. Compare the mathematical domain of $A(x)$ to the realistic domain in this context. Which choice correctly describes them?
7. A bacteria culture starts with 300 bacteria and grows by $18\%$ each hour. Create an exponential model for the population after $t$ hours and determine about how many hours it will take for the population to reach 2000 bacteria. Round to the nearest tenth of an hour.
8. A water tank starts with 200 liters and is filled at a constant rate of 15 liters per minute. Let $V$ be the volume (liters) after $t$ minutes. Write an equation for $V$ in terms of $t$, and choose appropriate axes labels and scale to graph the relationship for $0\le t\le 12$.
9. A circle has the equation $(x - 2)^2 + (y + 1)^2 = 9$. If this circle is translated so that its center moves to the origin, and then dilated by a factor of $\frac{2}{3}$ centered at the origin, what is the equation of the resulting circle?
10. What is the complex conjugate of $2-5i$?​
11. The table gives outputs of a function $f$ that has an inverse. Since points swap, $(x, f(x))$ on $f$ corresponds to $(f(x), x)$ on $f^{-1}$. Use the table to find $f^{-1}(13)$.

    $x$	$f(x)$
    2	1
    4	6
    6	10
    7	13
    9	18

12. The arithmetic sequence is $7, 11, 15, 19, \dots$ where the term number is $x$ (so $x=1$ gives 7). Write a linear function $f(x)$ that gives the $x$th term.
13. A store tracks the price of a collectible item over 4 weeks: Week 0: $100 Week 1: $110 Week 2: $120 Week 3: $130 Does this represent constant percent change (exponential) or constant additive change (linear)?
14. An investment of $1200 grows at 6% interest compounded annually. Set up an exponential equation modeling the balance after $t$ years and solve for how long it will take the investment to reach $2000. Let $t$ = number of years.
15. Let $z=7+0i$ (a real number). What is its complex conjugate $\overline{z}$?
16. Solve the system using elimination and give the exact solution. $\begin{cases} 5x - y = 9 \\ 2x + y = 12 \end{cases}$
17. A function $f$ is described as follows: it starts at the point $(0,2)$, increases until it reaches a relative maximum at $(2,6)$, then decreases crossing the $x$-axis at $(4,0)$, continues decreasing to a relative minimum at $(6,-4)$, and then increases thereafter. Which choice correctly gives the intervals where $f$ is increasing?
18. A company models profit by $P(x)=x^2-8x+15$, where $x$ is the number of items sold (in some unit). Factor $P(x)$ to find the break-even points (zeros of $P$).
19. Solve $2^x=3x+1$ using a table of values (successive approximations). Use the values below to choose the best approximation for the solution, to the nearest tenth. Table: - At $x=2.0$: $2^x=4.0$ and $3x+1=7.0$ - At $x=2.5$: $2^x\approx 5.7$ and $3x+1=8.5$ - At $x=2.8$: $2^x\approx 7.0$ and $3x+1=9.4$ - At $x=3.0$: $2^x=8.0$ and $3x+1=10.0$ - At $x=3.2$: $2^x\approx 9.2$ and $3x+1=10.6$ - At $x=3.4$: $2^x\approx 10.6$ and $3x+1=11.2$ - At $x=3.5$: $2^x\approx 11.3$ and $3x+1=11.5$ - At $x=3.6$: $2^x\approx 12.1$ and $3x+1=11.8$​
20. Are $f(x)=x^3-5$ and $g(x)=\sqrt[3]{x+5}$ inverse functions? Verify by computing $f(g(x))$ and $g(f(x))$ and checking whether both simplify to $x$.

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