Practice Test 51 - Algebra 2

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

A radioactive sample has a half-life of 50 years and starts with 200 g. Create an exponential model for the amount $N(t)$ after $t$ years, then solve for when 25 g remains. Round to the nearest year and interpret your result.
Compute $\overline{\left(\dfrac{2-i}{3+2i}\right)}$
Prove that any linear function $f(x)=mx+b$ grows by equal differences over equal intervals: show that for any real number $h$ , the difference $f(x+h)-f(x)$ is constant (independent of $x$ ). Which expression correctly simplifies $f(x+h)-f(x)$ ?
Multiply and write the result in standard form (polynomials are closed under multiplication): $(x^3 - 2x^2 + 3)(x - 5)$
Order the following by eventual growth rate (for sufficiently large $x$ ): linear $f(x)=100x$ , quadratic $g(x)=x^2$ , polynomial $h(x)=x^5$ , and exponential $p(x)=1.1^x$ .
A student writes a system of equations in matrix form as: $\begin{pmatrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ 0 & 5 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 7 \\ 2 \\ -1 \end{pmatrix}$ How many equations in this system have exactly two variables with non-zero coefficients?
The polynomial $g(x) = x^5 - 3x^4 - 4x^3 + 12x^2$ can be factored as $g(x) = x^2(x-2)^2(x+2)$ . Based on this factorization, which characteristic of the graph is most unusual or distinctive?
Add and write the result in standard form. This illustrates closure under addition: $\left(-2x^3+7x^2-x+1\right)+\left(5x^3-3x^2+4x-6\right).$
The arithmetic sequence is $14,\ 9,\ 4,\ -1,\ \dots$ with domain $n \in \{1,2,3,\dots\}$ . Write both a recursive formula and an explicit formula for $a_n$ .
A savings account starts with $2500$ and earns $5\%$ interest compounded annually. Create an exponential model for the balance after $t$ years, then determine how long it will take for the account to reach $4000$ . Round to the nearest tenth of a year and interpret your result in context.
The amount of a medication in a patient's bloodstream is modeled by $M(t)=240(0.85)^t$ , where $t$ is in hours and $M(t)$ is in milligrams. In this function, interpret the parameter $0.85$ in context.
On the complex plane (real axis horizontal, imaginary axis vertical), the complex number $z = -3 + 2i$ is represented by the point $(a,b)$ where $a$ is the real part and $b$ is the imaginary part. Which point should be plotted for $z$ ?
In the expression $2x(x-3)^2 + 5$ , which of the following correctly distinguishes the terms and the factors of the first term? First term: $2x(x-3)^2$
Two polynomial functions are given: $f(x) = x^4 - 5x^2 + 4$ and $g(x) = -x^4 + 5x^2 - 4$ . After factoring both functions completely, which statement correctly compares their graphical behaviors?
Function $f$ is given by $f(x)=\left(\dfrac{1}{2}\right)^x$ . Function $g$ is represented by the table: $\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline g(x) & 5 & 10 & 20 & 40 \end{array}$ Which function has the greater output at $x=3$ ?
Condense into a single logarithm (natural log): $\tfrac{3}{2}\ln(x)+\ln(y)-\tfrac{1}{2}\ln(z).$
A retirement account requires monthly payments. If the first payment is $800 and each subsequent payment increases by 2% from the previous payment, what is the total amount paid after 12 payments?
Show by composition whether $f(x)=x^3-5$ and $g(x)=\sqrt[3]{x+5}$ are inverse functions. Compute and simplify $f(g(x))$ and $g(f(x))$ .
For the quadratic equation $x^2+4x+4=0$ (with real coefficients), remember: every quadratic $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity. The discriminant $b^2-4ac$ determines whether the solutions are two distinct real ( $>0$ ), one repeated real (= $0$ , multiplicity 2), or two complex conjugates ( $<0$ ). Which statement correctly describes the solutions of $x^2+4x+4=0$ in the complex numbers?
Find the exact solution (as a logarithm) and an approximate solution for $t$ in $4\cdot e^{2t}=30$ . Use technology to evaluate the logarithm.

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