Practice Test 105 - Algebra 2

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

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1. Find the linear function $f(x)=mx+b$ with slope $m=-3$ that passes through the point $(2,5)$. What is $f(x)$?
2. A company's profit function is $P(x) = -2x^3 + 24x^2 - 72x + 50$, where $x$ represents the number of items produced (in hundreds) and $P(x)$ is profit in thousands of dollars. Analyzing the end behavior and critical points, which statement best describes the company's long-term production strategy?
3. Solve for $x$ using the inverse relationship: $\log_2(x)=6$.​
4. Graph the polynomial function $p(x)=(x+2)^2(x-1)(x-3)$ showing all zeros (with whether the graph crosses or touches at each zero) and the correct end behavior.​
5. Determine whether the system has no solution, one solution, or infinitely many solutions: $\begin{cases} 2x + y = 5\\ 4x + 2y = 10 \end{cases}$
6. A shipping company models the cost (in dollars) to ship a package with weight $w$ (in pounds) by $C(w)=5+2\sqrt{w-1}$. In this context, packages must weigh at least 1 pound. What is the realistic domain for $w$?
7. Solve for $x$: $\log_2(x)=4$.
8. A student graphs two equations and observes that the lines are parallel. Which of the following systems could produce this result?
9. Use the key conjugate property $(a+bi)(a-bi)=a^2+b^2$ to compute $(4+3i)(4-3i)$. (This product is always real and equals the modulus squared.)
10. A town's population is recorded every year. Year $y$: 0, 1, 2, 3 Population $P(y)$: 15000, 15600, 16224, 16872.96 Does the population show constant percent change? If so, what is the percent growth rate per year?​​
11. Use synthetic division to find the quotient and remainder when $P(x)=2x^4+x^3-3x^2+4x-1$ is divided by $(x-2).$ Set up with $c=2$ and coefficients $2,1,-3,4,-1$. Read the quotient coefficients from the bottom row and the remainder as the last value (which equals $P(2)$). Which is correct?
12. Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$.​
13. Sound intensity level in decibels is modeled by $\text{dB}=10\log_{10}\left(\frac{I}{I_0}\right)$, where $I_0$ is a reference intensity. A sound has intensity $5000I_0$. Create an equation and find its decibel level. Round to the nearest tenth of a decibel and interpret the result.
14. A town's population is 18,000 and grows by 4% each year. Which function models the population after $t$ years (with $t=0$ now)?
15. The solutions to the equation $x^2 + bx + c = 0$ are $x = 3 \pm 2i$. What are the values of $b$ and $c$?
16. Find the modulus $r$ and a correct argument $\theta$ (in degrees) for $z=-\sqrt{3}-i$, then write $z$ in polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with quadrant adjustment.
17. The graphs of $y=f(x)$ and $y=g(x)$ intersect at the point $(a,b)$. Which statement must be true and explains why the solution(s) to $f(x)=g(x)$ come from intersection points?
18. A local theater charges a fixed rental fee plus an additional cost per hour for use of their facility. The total cost for renting the theater for 3 hours is $450, and for 7 hours is $750. Which equation represents the relationship between the total cost $C$ (in dollars) and the number of hours $h$?
19. What are $q(x)$ and $r(x)$ when $\frac{x^4-2x^2+x+1}{x^2+1}$ is written as $q(x)+\frac{r(x)}{x^2+1}$ with $\deg(r)<2$ (as required by the polynomial division algorithm)?
20. Express $\frac{x^3-4x^2+x+6}{x-3}$ as $q(x)+\frac{r(x)}{x-3}$ using the polynomial division algorithm, with $\deg(r)<\deg(x-3)$.

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