Practice Test 23 - Algebra 2

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

A sequence is a function defined only at integer inputs. Which set could be the domain of a sequence $a(n)$ (viewed as a discrete function)?
Graph $f(x)=\dfrac{(x-3)(x+2)}{(x-1)(x+4)}$ by identifying its zeros, vertical asymptotes, and horizontal asymptote.
Use technology to find $A^{-1}$ and solve $AX=B$ by computing $X=A^{-1}B$ . $A=\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix},\quad B=\begin{bmatrix}6\\5\\3\end{bmatrix}$ What is $X$ ?
A delivery company tracks total miles driven each day for a week. The sequence begins 120, 135, 150, 165 (miles). Is the change best described as exponential growth/decay or neither (linear/additive), and why?
Compute using conjugates and express in $a+bi$ form: $\frac{1+2i}{3-i}.$
A student claims that for any polynomials $P(x)$ and $Q(x)$ , the expression $(P(x) + Q(x))^2$ is always equal to $P(x)^2 + Q(x)^2$ . Which pair of simple polynomials provides the smallest counterexample to disprove this claim?
A town's population is modeled by $P(t) = 18{,}000(1.03)^t$ , where $t$ is the number of years since 2026. In the function $P(t) = 18{,}000(1.03)^t$ , what does the base $1.03$ represent?
A bank account earns interest such that the balance after $t$ years is given by $B(t) = 5000(1.045)^t$ . A savings bond increases in value according to $V(t) = 3000 + 180t$ . Which statement best describes the growth patterns of these two investments?
Which quadratic equation is most efficiently solved by factoring?
Describe the end behavior of the polynomial $h(x)= -3x^5+2x^3-7x+1$ as $x\to\pm\infty$ .
Let $f(x)=x^3$ (an odd-symmetric cubic through the origin). What transformation changes $f(x)$ to $g(x)=-\tfrac{1}{2}f(x)$ ?
Solve the system using substitution: $\begin{cases} y = x - 5\\ 2x + y = 7 \end{cases}$
Apply the sum of cubes identity to factor completely: $x^3 + 8.$
All points $(x,y)$ that are 5 units from the point $(1,2)$ form a circle. Write the equation of this circle in standard form. (Use the Pythagorean Theorem/distance formula.)
A certain chemical concentration starts at 80 units and increases by $6\%$ each day. Create an exponential model for the concentration $C(t)$ after $t$ days, then solve for when it reaches 120 units. Round to the nearest tenth of a day and interpret your result.
Prove the identity $(a - b)^3 = a^3 - 3a^2 b + 3 a b^2 - b^3$ by expanding $(a - b)^3$ , then apply it to expand $(x - 2)^3$ .
Compare the growth of these functions for large $x$ : $f(x)=3x$ (linear), $g(x)=x^2$ (quadratic), $h(x)=x^5$ (polynomial), and $p(x)=1.5^x$ (exponential). Which function eventually grows fastest?
A landscaper uses the model $L(x)=\sqrt{x-2}$ to estimate the length (in meters) of a garden border after $x$ meters of fencing have been laid out. In this context, $x$ cannot exceed 50 meters. What is the appropriate realistic domain for $L$ ?
A $3 \times 3$ system is given in matrix form. Use technology to find $A^{-1}$ and then compute $X = A^{-1} B$ . $A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 2 & 1 \\ 3 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 8 \\ 11 \end{bmatrix}$ What is $X$ ?
A point on the complex plane is $(a,b)$ , representing $z=a+bi$ . Which complex number has modulus $5$ and lies in Quadrant II (negative real part, positive imaginary part)?

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