Practice Test 88 - Algebra 2

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

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1. Solve for $t$ and give both exact and approximate forms: $9 \cdot e^{0.5t} = 120$ (use technology to evaluate the logarithm).
2. A geometric sequence is given: $3,\ 6,\ 12,\ 24,\dots$ Let $g(n)$ be the $n$th term with $g(1)=3$. Which exponential function gives $g(n)$?
3. Rationalize the denominator and write in standard form $a+bi$: $\frac{2-i}{3+2i}$
4. For the polynomial $q(x) = -x^5 + 4x^3 - x,$ describe the end behavior as $x \to \pm \infty$.
5. The height of a tossed ball (in meters) after $t$ seconds is modeled by $h(t)=-5t^2+20t+1$. Identify the y-intercept of $h$ and interpret it in context.
6. A savings account is modeled by the compound interest expression $A = P(1+r)^n$, where $P$ is the initial deposit, $r$ is the interest rate per period, and $n$ is the number of periods. Which interpretation correctly chunks the expression to show what part does not depend on $P$?
7. Simplify using the inverse relationship between exponentials and logarithms: $\log_3(3^8)$.​
8. Solve the system by any method. Then interpret the result: do the lines intersect, are they parallel, or are they the same line? $\begin{cases} 3x + 2y = 8 \\ 6x + 4y = 16 \end{cases}$
9. For $f(x)=(x+2)^2$ with domain all real numbers, a horizontal line can intersect the graph twice, so $f$ is not invertible. Which restriction makes $f$ pass the horizontal line test?
10. Points $z_1$ and $z_2$ on the complex plane form a line segment. The midpoint is $\dfrac{z_1+z_2}{2}$. What complex number is the midpoint of the segment joining $z_1=-6+4i$ and $z_2=2-2i$?
11. Graph the rational function $f(x)=\frac{(x-2)(x+3)}{(x-5)(x+1)}$ identifying its zeros, vertical asymptotes, and horizontal asymptote (end behavior). Choose the option that lists the correct features and matches a reasonable sketch description.
12. Let $z_1 = 8 - 4i$ and $z_2 = 3 + 9i$. What is $z_1 - z_2$ in standard form $a + bi$?
13. Let $z=2-5i$. Which statement correctly describes the geometric relationship between $z$ and its conjugate $\overline{z}$ on the complex plane?​
14. Recognize structure and factor. The expression $8x^3+27$ is a sum of cubes. Which factorization is correct?
15. A delivery company charges $30 per hour for a driver plus $2 per mile traveled. Let $h$ be hours and $m$ be miles. 1) Create an equation for total cost $C$ (dollars) in terms of $h$ and $m$. 2) To graph on a 2D coordinate plane, you decide to fix $h=3$ hours and graph cost as a function of miles $m$ for $0\le m\le 100$. Which choice gives the correct equation and a correct graph setup for the $h=3$ case?
16. When $(x - y)^7$ is expanded, how many terms have odd coefficients?
17. Solve and evaluate using technology: $7\cdot 2^{t/4}=60$. Give $t$ in exact logarithmic form and as a decimal approximation.
18. On the complex plane, let $z_1=3+2i$ and $z_2=1-i$. Using the distance interpretation $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (the same as the coordinate distance formula between $(a,b)$ and $(c,d)$), what is the distance between $z_1$ and $z_2$?
19. In finance, compound growth can be modeled by $A=P(1+r)^t$, where $A$ is the final amount, $P$ is the initial principal, $r$ is the rate per period, and $t$ is the number of periods. Solve $A=P(1+r)^t$ for $P$.
20. Describe the end behavior by finding the horizontal asymptote of $f(x)=\frac{3x^2-x+7}{2x^2+5x-1}.$

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