Practice Test 77 - Algebra 2

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

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1. Identify the amplitude, period, and midline of the trigonometric function $f(x)=3\sin(2x)-1$.
2. In the expression $3(2x+1)^2$, what is the coefficient of $(2x+1)^2$?​
3. A car travels at a steady speed of 55 miles per hour for $t$ hours. Is the rate of change of distance with respect to time constant? What type of function models distance vs. time?
4. For the sequence defined explicitly by $a_n=9\left(\tfrac{1}{2}\right)^{n-1}$ for integers $n\ge 1$, write the equivalent recursive definition.
5. Write the system as a matrix equation $AX=B$ (variable order $x, y, z$). Remember to use a coefficient of $0$ for any missing variable. $\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$​
6. Consider the matrix equation $A\vec{x} = \vec{b}$ where $A = \begin{pmatrix} 1 & 2 & -1 \\ 3 & 1 & 2 \\ 2 & -1 & 1 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 5 \\ 8 \\ 3 \end{pmatrix}$. After computing $A^{-1}$, a student finds that $\vec{x} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$. However, when checking by substitution, the third equation gives $2(2) + (-1)(1) + 1(-1) = 2$, not $3$. What is the most likely error?
7. Factor the quadratic function $f(x)=x^2+7x+12$. Then use the zero product property to find the zeros and identify the x-intercepts of the graph.
8. A bike rental shop charges a one-time sign-up fee of $12 plus $4 per hour of rental. Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$, and choose appropriate axes labels and a reasonable scale to graph the relationship for $0\le h\le 10$.
9. Which function eventually grows fastest as $x\to\infty$?
10. For the rational function $f(x)=\frac{(x-2)(x+1)}{(x-4)(x+3)},$ identify the zeros, vertical asymptotes, and horizontal asymptote (end behavior).​
11. Find the sum of the first $6$ terms of the geometric series $3+6+12+24+48+96.$
12. Determine whether the data are best modeled by a linear or exponential function, then write the function. $x$: 1, 2, 3, 4 $y$: 2, 6, 18, 54
13. Polynomial identities that hold for real numbers also hold for complex numbers because addition and multiplication work the same way in $\mathbb{C}$. Use the identity $(a+b)^2=a^2+2ab+b^2$ to verify it for the complex numbers $a=1+i$ and $b=2-i$. Which option correctly shows that both sides simplify to the same complex number?
14. Which expression is always real and nonnegative for real numbers $a$ and $b$ (not both zero), and equals the modulus squared $|a+bi|^2$? Choose the correct expression.
15. Write the system as a matrix equation $AX=B$ (use variable order $x, y, z$). Be sure to include a $0$ coefficient for any missing variable: $\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$
16. On the complex plane (real axis horizontal, imaginary axis vertical), the complex number $z = -3 + 4i$ is represented by the point $(a,b)$ where $z=a+bi$. What is the modulus $|z|$ (the distance from the origin), and which point represents $z$?
17. A rectangle has length $(x+5)$ meters and width $(x-2)$ meters. Create an equation in two variables for the area $A$ (in square meters) in terms of $x$, then select appropriate axes labels and a reasonable scale to graph $A$ versus $x$ for $2 \le x \le 10$.
18. Find the center and radius of the circle $x^2 + y^2 - 8x - 6y + 9 = 0$.
19. Consider $K(x)=\frac{(x+4)^2}{2x-3}.$ Interpret by chunking $(x+4)^2$ as one entity and $(2x-3)$ as another. Which statement correctly describes the structure and dependence on $x$?
20. For the rational function $f(x)=\dfrac{x^2-9}{x^2-4}$, identify the zeros, vertical asymptotes, and horizontal asymptote.​

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