Practice Test 50 - Algebra 2

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

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1. Consider $g(x)=x^2(x+1)-4(x+1).$ If you view $(x+1)$ as a single entity, which statement best describes the structure you can see within the complexity?
2. Translate the recursive definition to an explicit formula (with integer domain $n\ge 1$): Recursive: $a_1=6,\ a_{n+1}=3a_n$. Which explicit formula matches this sequence?
3. An exponential function has the form $g(x)=a\cdot b^x$ and passes through $(0,6)$ and $(2,54)$. Which function is correct?
4. The inverse of a matrix is given. Use the inverse-matrix method to solve $AX=B$ by computing $X=A^{-1}B$. $A^{-1}=\begin{bmatrix}1&-2\0&1\end{bmatrix},\quad B=\begin{bmatrix}5\3\end{bmatrix}$ What is $X$?
5. Using the table shown, determine which system of equations has the solution that corresponds to the intersection point of Line A and Line B.
6. Which matrix equation $AX=B$ correctly represents the system (use variable order $x, y$)? $$\begin{cases} -2x+7y=4\\ 5x+y=9 \end{cases}$$
7. Sketch $f(x)=\frac{x^2-1}{x^2-4}$ showing all vertical asymptotes, all zeros, and the horizontal asymptote. Which set of features is correct?
8. The temperature deviation from average (in °C) during a day is modeled by $T(t)=3\cos\left(\frac{\pi}{12}t\right)$, where $t$ is hours after midnight. What is the period of $T(t)$, and what does it mean in context?
9. A town's population is modeled by the function $P(t)=48{,}000(1.06)^t$, where $t$ is in years. Identify the constant percent rate and classify the change.
10. A ride's vertical position (in meters) is modeled by $y(t)=2\cos\left(\frac{\pi}{4}t\right)$, where $t$ is time in seconds. On what interval(s) is $y(t)$ positive for $0\le t\le 8$?
11. Find the inverse function $f^{-1}(x)$ for $f(x)=2x^3$.​​
12. The graph shows function $r(x)$. Based on the graph, what is $r^{-1}(0) + r^{-1}(2)$?
13. In the rational expression $\dfrac{3x^2 - 2x + 5}{x - 1}$, what are the terms in the numerator?
14. Two linear equations are given: $3x - 2y = 7$ and $6x + ky = 14$. For what value of $k$ will this system have infinitely many solutions?
15. What are the center and radius of the circle in standard form $ $(x+7)^2$ + $(y-3)^2$ = 64 $?
16. Which equation is most efficient to solve by factoring (rather than the quadratic formula)?
17. Solve using the method of completing the square: $x^2 + 6x - 2 = 0.$ First rewrite it in the form $(x - p)^2 = q$.
18. Derive the finite geometric series sum formula for $r \ne 1$. Start with $S_n = a + ar + ar^2 + \cdots + ar^{n-1}.$ Multiply both sides by $r$ and subtract to show the telescoping cancellation, then solve for $S_n$. Which expression is correct?
19. Write matrices $A$, $X$, and $B$ for the system (use variable order $x, y, z$): $$\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$$ (Remember to use 0 for any missing variable in an equation.)
20. A culture of bacteria starts with 500 cells and grows by 12% each hour. 1) Write an equation for the population $P$ in terms of time $t$ (hours). 2) Choose appropriate axes labels and a reasonable scale to graph the model for $0\le t\le 10$. Which choice is correct?

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