Practice Test 32 - Algebra 2

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

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1. A graphing calculator shows that $h(x) = e^{x-2}$ and $k(x) = \frac{6}{x+1}$ intersect at approximately $x = 1.5$. To verify this intersection point, which calculation should produce a result closest to zero?
2. The amount of a medication in the bloodstream is modeled by $M(h)=120(0.88)^h$, where $h$ is the number of hours after the dose. What does the base $0.88$ represent in context?
3. Convert from polar to rectangular form using $a=r\cos\theta$ and $b=r\sin\theta$: $5\left(\cos 45^\circ + i\sin 45^\circ\right).$
4. Solve the exponential equation $3\cdot 10^{2t}=75$ for $t$. Give the exact solution (as a logarithm) and then approximate $t$ using technology (calculator).
5. For $p(x)=(x-1)^4(x+3),$ identify zeros and determine crossing vs touching at each zero. Also determine the end behavior.
6. The complex conjugate of $a+bi$ is $a-bi$ (flip the sign of the imaginary part only). What is the conjugate of $3-4i$?
7. Given the explicit formula $a_n=7\cdot 2^{n-1}$ for $n\ge 1$, write the equivalent recursive definition.
8. Which statement correctly applies the Rational Zeros Theorem to $P(x)=6x^3-x^2+4x-5?$
9. What is the coefficient of $x^2$ in the polynomial $3x^3 - 7x^2 + 2x - 9$?
10. Let $f(x)=x^3$ and $g(x)=4x$. Use the intersections of $y=f(x)$ and $y=g(x)$ to solve $f(x)=g(x)$ (exact solutions).
11. A sequence is defined recursively by $a_1=5$ and $a_{n+1}=a_n+7$ for $n \in \{1,2,3,\dots\}$. Write the equivalent explicit formula for $a_n$.
12. Find all intersection point(s) of the line and the circle (solve the system). You may solve algebraically and confirm graphically. $\begin{cases} x^2 + y^2 = 25 \\ y = -3x \end{cases}$ What are the solution(s) $(x,y)$?
13. Let $a$ be a sequence (discrete function) with domain $n\in\{1,2,3,\dots\}$ defined by $a_1=5$ and $a_{n+1}=a_n+7$ for $n\ge 1$. What is $a_5$?
14. A complex number $w=-1-4i$ is plotted on the complex plane with coordinates $(a,b)$ corresponding to $a+bi$. What are the coordinates of the point for $w$, and what is the modulus $|w|=\sqrt{a^2+b^2}$?
15. Transform and solve by completing the square: $x^2+10x-3=0.$ Write it as $(x-p)^2=q$ and then solve for $x$.
16. A formula used in some models of electrical power is $P=\dfrac{V^2}{R}$, where $P$ is power, $V$ is voltage, and $R$ is resistance. Solve for $V$ in terms of $P$ and $R$.
17. A student claims the system (variable order $x, y$) $\begin{cases} 6x-y=2\\ -3x+5y=11 \end{cases}$ can be written as $\begin{bmatrix}6&-1\\-3&5\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\11\end{bmatrix}.$ Which statement best verifies the claim?
18. On the complex plane, the point $z=a+bi$ corresponds to $(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (the same as the coordinate geometry distance formula). What is the distance between $z_1=3+2i$ and $z_2=1-i$?
19. Use the identity $x^2 + 2xy + y^2 = (x + y)^2$ to factor the polynomial $9m^2 + 12mn + 4n^2.$
20. A regular hexagon in the complex plane has its center at $2 + 3i$ and one vertex at $5 + 3i$. What is the distance between two adjacent vertices of this hexagon?

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