Practice Test 71 - Algebra 2

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

Use the inverse property $b^{\log_b(x)}=x$ (for $x>0$ ) to simplify $2^{\log_2(17)}$ .
In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$ , initial speed $u$ , acceleration $a$ , and displacement $s$ . Solve $v^2 = u^2 + 2as$ for $a$ (treat the other variables as constants, using the same inverse-operation reasoning as solving a numeric equation).
A solution's pH is given by $\text{pH}=-\log([H^+])$ (base 10). If $[H^+]=10^{-7}$ , what is the pH? Use log properties.
A student graphs $y = |x - 3| + 1$ and $y = \frac{1}{2}x + 4$ on the same coordinate plane. Based on the intersection points of these graphs, which equation below would have the same solutions?
The graph shows the voltage function $V(t) = 12\cos(120\pi t + \frac{\pi}{3})$ for an alternating current circuit. Which statement correctly interprets the phase shift and its effect on the zero crossings compared to the standard cosine function?
A company analyzes three pricing strategies over time: Strategy 1 gives profit $P_1(t) = 2^t - 1$ thousand dollars, Strategy 2 gives $P_2(t) = t^3 + 10t$ thousand dollars, and Strategy 3 gives $P_3(t) = 15t + 5$ thousand dollars, where $t$ is time in months. At $t = 4$ , Strategy 2 yields the highest profit. What is the most likely scenario at $t = 15$ ?
The decibel level of a sound can be modeled by $\text{dB}=10\log_{10}(I/I_0)$ , where $I$ is the sound intensity and $I_0$ is a reference intensity. A sound has intensity $500$ times the reference (so $I/I_0=500$ ). Find the decibel level to the nearest tenth and interpret your result.
The sum/difference of cubes identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ can be used with complex numbers. Let $a=1+i$ and $b=1-i$ . What is $a^3-b^3$ ?
Recognize the pattern and factor completely (sum of cubes): $8x^3+27$
Let $f(x)=|x|$ . The graph of $g$ is shown along with $f$ . The function $g$ is of the form $g(x)=f(x)+k.$ What is the value of $k$ ? (Use the graph: $f$ has vertex at $(0,0)$ and $g$ has vertex at $(0,3)$ .)
Consider the expression $3x^2(2x - 5) + 7x - 4$ . When this expression is written in standard form, how many terms contain $x$ as a factor?
Use polynomial division to write $\dfrac{x^4-2x^3+3x^2+4}{x^2+1}$ in the form $q(x)+\dfrac{r(x)}{x^2+1}$ , where $\deg(r)<2$ .
Add and express in standard form $a+bi$ : $(5+3i)+(2-7i).$
For the geometric series $5+15+45+135+405$ , identify $a$ , $r$ , and $n$ , then find $S_n$ (the sum of all terms shown).
What is the degree of the polynomial product? (This uses the fact that polynomials are closed under multiplication and that degrees add when multiplying.) If $P(x)$ has degree $3$ and $Q(x)$ has degree $4$ , what is the degree of $P(x)\cdot Q(x)$ ?
The polynomial $h(x) = kx^4 - 8x^3 + mx^2 + nx - 6$ has $\frac{3}{2}$ as a rational zero. If all coefficients are integers and the Rational Zero Theorem is to be applied effectively, what constraint must $k$ satisfy?
A school is buying notebooks and folders. Let $x$ = number of notebooks and $y$ = number of folders. Constraints: - Notebooks cost $2 each and folders cost $3 each; the budget is at most $60: $2x+3y\le 60$ . - They need at least 10 notebooks: $x\ge 10$ . - They need at least 8 folders: $y\ge 8$ . - Items must be whole numbers and non-negative. Which system correctly represents all constraints?
A ball is thrown upward from a platform. Its height (in meters) after $t$ seconds is modeled by $h(t)=-5(t-2)^2+20$ for $0\le t\le 5$ . On what interval is $h(t)$ increasing?
A cylindrical tank has volume $V=\pi r^2 h$ , where $r$ is radius (meters) and $h$ is height (meters). If the height is fixed at $h=3$ meters, what two-variable equation relates $V$ and $r$ , and what axes labels and a reasonable scale would you use to graph it for $0\le r\le 5$ ?
Consider the polynomial function $f(x) = -2x^4 + 8x^3 - 6x^2$ . After factoring completely, which statement correctly describes both the zeros and the end behavior of this function?

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